FormulasFoundationfoundation_paper_3_quantitative_aptitudeChapter 6
Back to Chapters

Chapter 6

Topics and formulas

Filter by Topic

All Formulas

41 formulas

Nth Term of an Arithmetic Progression (AP)

Topic: Logical ReasoningChapter: Chapter 6
an=a+(n−1)da_n = a + (n-1)d
This formula calculates the nth term of an AP. 'a_n' is the nth term, 'a' is the first term, 'n' is the number of terms, and 'd' is the common difference between consecutive terms. Use this when you need to find a specific term in an arithmetic sequence without listing all preceding terms.

Sum of N Terms of an Arithmetic Progression (AP)

Topic: Logical ReasoningChapter: Chapter 6
Sn=n2[2a+(n−1)d]S_n = \frac{n}{2}[2a + (n-1)d]
This formula calculates the sum of the first 'n' terms of an AP. 'S_n' is the sum of n terms, 'a' is the first term, 'n' is the number of terms, and 'd' is the common difference. An alternative form is Sn=n2(a+l)S_n = \frac{n}{2}(a + l), where 'l' is the last term. Use this to quickly find the total sum of a given number of terms in an AP.

Arithmetic Mean (AM)

Topic: Logical ReasoningChapter: Chapter 6
AM=a+b2AM = \frac{a+b}{2}
The arithmetic mean of two numbers 'a' and 'b' is their sum divided by two. For a series of numbers, it's the sum of all numbers divided by the count of numbers. It represents the central value of a set of numbers.

Nth Term of a Geometric Progression (GP)

Topic: Logical ReasoningChapter: Chapter 6
an=arn−1a_n = ar^{n-1}
This formula calculates the nth term of a GP. 'a_n' is the nth term, 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms. Use this when you need to find a specific term in a geometric sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number.

Sum of N Terms of a Geometric Progression (GP)

Topic: Logical ReasoningChapter: Chapter 6
Sn=a(rn−1)r−1 (if r > 1)orSn=a(1−rn)1−r (if r < 1)S_n = \frac{a(r^n - 1)}{r-1} \text{ (if r > 1)} \quad \text{or} \quad S_n = \frac{a(1 - r^n)}{1-r} \text{ (if r < 1)}
This formula calculates the sum of the first 'n' terms of a GP. 'S_n' is the sum of n terms, 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms. The choice of formula depends on whether the common ratio 'r' is greater than or less than 1. If r=1, then Sn=naS_n = na. Use this to find the total sum of a given number of terms in a GP.

Sum of Infinite Terms of a Geometric Progression (GP)

Topic: Logical ReasoningChapter: Chapter 6
S∞=a1−rS_\infty = \frac{a}{1-r}
This formula calculates the sum of an infinite GP, but only if the absolute value of the common ratio 'r' is less than 1 (i.e., |r| < 1). 'S_\infty' is the sum to infinity, 'a' is the first term, and 'r' is the common ratio. If |r| ≥\ge 1, the sum to infinity does not converge.

Geometric Mean (GM)

Topic: Logical ReasoningChapter: Chapter 6
GM=abGM = \sqrt{ab}
The geometric mean of two positive numbers 'a' and 'b' is the square root of their product. For 'n' positive numbers, it's the nth root of their product. It is often used when dealing with growth rates or ratios.

Sum of n terms of an Arithmetic Progression (AP) - Alternative

Topic: Logical ReasoningChapter: Chapter 6
Sn=n2(a+l)S_n = \frac{n}{2} (a + l)
An alternative formula to calculate the sum of the first n terms of an Arithmetic Progression when the first term (a) and the last term (l) are known. Here, S_n is the sum of n terms, n is the number of terms, a is the first term, and l is the last term.

Sum of n terms of a Geometric Progression (GP) - when r > 1

Topic: Logical ReasoningChapter: Chapter 6
Sn=a(rn−1)r−1S_n = \frac{a(r^n - 1)}{r - 1}
Calculates the sum of the first n terms of a Geometric Progression when the common ratio (r) is greater than 1. Here, S_n is the sum of n terms, a is the first term, n is the number of terms, and r is the common ratio.

Sum of n terms of a Geometric Progression (GP) - when r < 1

Topic: Logical ReasoningChapter: Chapter 6
Sn=a(1−rn)1−rS_n = \frac{a(1 - r^n)}{1 - r}
Calculates the sum of the first n terms of a Geometric Progression when the common ratio (r) is less than 1. Here, S_n is the sum of n terms, a is the first term, n is the number of terms, and r is the common ratio.

nth Term of an Arithmetic Progression (AP)

Topic: Arithmetic and Geometric ProgressionsChapter: Chapter 6
an=a+(n−1)da_n = a + (n-1)d
This formula calculates the n-th term (a_n) of an arithmetic progression, where 'a' is the first term, 'n' is the term number, and 'd' is the common difference between consecutive terms.

Sum of the first n Terms of an Arithmetic Progression (AP)

Topic: Arithmetic and Geometric ProgressionsChapter: Chapter 6
Sn=n2[2a+(n−1)d]orSn=n2[a+an]S_n = \frac{n}{2}[2a + (n-1)d] \quad \text{or} \quad S_n = \frac{n}{2}[a + a_n]
This formula calculates the sum (S_n) of the first 'n' terms of an arithmetic progression. 'a' is the first term, 'n' is the number of terms, 'd' is the common difference, and 'a_n' is the n-th term. The second form is used when the last term is known.

Arithmetic Mean (AM)

Topic: Arithmetic and Geometric ProgressionsChapter: Chapter 6
AM=a+b2AM = \frac{a+b}{2}
The arithmetic mean of two numbers 'a' and 'b' is the value that, when inserted between them, forms an arithmetic progression. For a set of 'n' numbers, it's the sum of all numbers divided by their count.

nth Term of a Geometric Progression (GP)

Topic: Arithmetic and Geometric ProgressionsChapter: Chapter 6
an=arn−1a_n = ar^{n-1}
This formula calculates the n-th term (a_n) of a geometric progression, where 'a' is the first term, 'n' is the term number, and 'r' is the common ratio between consecutive terms.

Sum of the first n Terms of a Geometric Progression (GP)

Topic: Arithmetic and Geometric ProgressionsChapter: Chapter 6
Sn=a(rn−1)r−1for req1,r>1orSn=a(1−rn)1−rfor req1,r<1S_n = a\frac{(r^n - 1)}{r-1} \quad \text{for } r eq 1, r > 1 \quad \text{or} \quad S_n = a\frac{(1 - r^n)}{1-r} \quad \text{for } r eq 1, r < 1
This formula calculates the sum (S_n) of the first 'n' terms of a geometric progression. 'a' is the first term, 'n' is the number of terms, and 'r' is the common ratio. The choice of formula depends on whether 'r' is greater or less than 1.

Sum of an Infinite Geometric Progression (GP)

Topic: Arithmetic and Geometric ProgressionsChapter: Chapter 6
S∞=a1−rfor ∣r∣<1S_\infty = \frac{a}{1-r} \quad \text{for } |r| < 1
This formula calculates the sum (S_infinity) of an infinite geometric progression, provided that the absolute value of the common ratio 'r' is less than 1. 'a' is the first term. If |r| \ge 1, the sum does not converge.

Geometric Mean (GM)

Topic: Arithmetic and Geometric ProgressionsChapter: Chapter 6
GM=abGM = \sqrt{ab}
The geometric mean of two positive numbers 'a' and 'b' is the value that, when inserted between them, forms a geometric progression. For 'n' positive numbers, it's the n-th root of their product.

Sum of N Terms of an Arithmetic Progression (AP)

Topic: Arithmetic and Geometric ProgressionsChapter: Chapter 6
Sn=n2[2a+(n−1)d]S_n = \frac{n}{2} [2a + (n-1)d]
Calculates the sum of the first 'n' terms of an Arithmetic Progression. 'S_n' is the sum, 'a' is the first term, 'n' is the number of terms, and 'd' is the common difference.

Sum of N Terms of an Arithmetic Progression (AP) - Alternative

Topic: Arithmetic and Geometric ProgressionsChapter: Chapter 6
Sn=n2(a+l)S_n = \frac{n}{2} (a + l)
Calculates the sum of the first 'n' terms of an Arithmetic Progression when the first term 'a' and the last term 'l' (which is the nth term) are known. 'S_n' is the sum, 'n' is the number of terms.

Common Difference of an Arithmetic Progression (AP)

Topic: Arithmetic and Geometric ProgressionsChapter: Chapter 6
d=an−an−1d = a_n - a_{n-1}
Calculates the common difference 'd' between consecutive terms in an Arithmetic Progression. 'a_n' is the nth term and 'a_{n-1}' is the (n-1)th term.

Arithmetic Mean (AM) of Two Numbers

Topic: Arithmetic and Geometric ProgressionsChapter: Chapter 6
AM=a+b2AM = \frac{a+b}{2}
Calculates the Arithmetic Mean of two numbers 'a' and 'b'. This is the middle term if 'a', AM, 'b' are in AP.

Sum of N Terms of a Geometric Progression (GP) when r > 1

Topic: Arithmetic and Geometric ProgressionsChapter: Chapter 6
Sn=a(rn−1)r−1S_n = \frac{a(r^n - 1)}{r - 1}
Calculates the sum of the first 'n' terms of a Geometric Progression when the common ratio 'r' is greater than 1. 'S_n' is the sum, 'a' is the first term, and 'n' is the number of terms.

Sum of N Terms of a Geometric Progression (GP) when r < 1

Topic: Arithmetic and Geometric ProgressionsChapter: Chapter 6
Sn=a(1−rn)1−rS_n = \frac{a(1 - r^n)}{1 - r}
Calculates the sum of the first 'n' terms of a Geometric Progression when the common ratio 'r' is less than 1. 'S_n' is the sum, 'a' is the first term, and 'n' is the number of terms.

Sum of Infinite Terms of a Geometric Progression (GP)

Topic: Arithmetic and Geometric ProgressionsChapter: Chapter 6
S∞=a1−rS_{\infty} = \frac{a}{1 - r}
Calculates the sum of an infinite Geometric Progression, which converges only when the absolute value of the common ratio '|r|' is less than 1. 'S_\infty' is the sum, 'a' is the first term.

Common Ratio of a Geometric Progression (GP)

Topic: Arithmetic and Geometric ProgressionsChapter: Chapter 6
r=anan−1r = \frac{a_n}{a_{n-1}}
Calculates the common ratio 'r' between consecutive terms in a Geometric Progression. 'a_n' is the nth term and 'a_{n-1}' is the (n-1)th term.

Geometric Mean (GM) of Two Numbers

Topic: Arithmetic and Geometric ProgressionsChapter: Chapter 6
GM=abGM = \sqrt{ab}
Calculates the Geometric Mean of two positive numbers 'a' and 'b'. This is the middle term if 'a', GM, 'b' are in GP.

nth Term of an Arithmetic Progression (AP)

Topic: Time Value of MoneyChapter: Chapter 6
an=a+(n−1)da_n = a + (n-1)d
This formula calculates the value of the nth term (a_n) in an Arithmetic Progression. Here, a represents the first term, n is the number of terms, and d is the common difference between consecutive terms. It's used when cash flows increase or decrease by a fixed amount each period.

Sum of n Terms of an Arithmetic Progression (AP)

Topic: Time Value of MoneyChapter: Chapter 6
Sn=n2[2a+(n−1)d]S_n = \frac{n}{2}[2a + (n-1)d]
This formula calculates the sum (S_n) of the first n terms of an Arithmetic Progression. Alternatively, it can be expressed as S_n = \frac{n}{2}[a + a_n]. It's applied to sum up a series of cash flows that change by a constant amount, such as certain types of growing or declining annuities.

nth Term of a Geometric Progression (GP)

Topic: Time Value of MoneyChapter: Chapter 6
an=arn−1a_n = ar^{n-1}
This formula determines the value of the nth term (a_n) in a Geometric Progression. Here, a is the first term, n is the number of terms, and r is the common ratio. This is useful when cash flows grow or shrink by a constant percentage each period.

Sum of n Terms of a Geometric Progression (GP)

Topic: Time Value of MoneyChapter: Chapter 6
Sn=a(rn−1)r−1if r>1orSn=a(1−rn)1−rif r<1S_n = a\frac{(r^n - 1)}{r-1} \quad \text{if } r > 1 \quad \text{or} \quad S_n = a\frac{(1 - r^n)}{1-r} \quad \text{if } r < 1
This formula calculates the sum (S_n) of the first n terms of a Geometric Progression. a is the first term, n is the number of terms, and r is the common ratio. This is crucial for calculating the future value or present value of annuities where cash flows grow or shrink at a constant rate (e.g., growing annuities).

Sum of Infinite Terms of a Geometric Progression (GP) (Perpetuity Formula)

Topic: Time Value of MoneyChapter: Chapter 6
S∞=a1−rif ∣r∣<1S_\infty = \frac{a}{1-r} \quad \text{if } |r| < 1
This formula calculates the sum (S_\infty) of an infinite number of terms in a Geometric Progression, provided the absolute value of the common ratio (|r|) is less than 1. a is the first term. This is directly applicable to valuing perpetuities, which are infinite streams of constant or growing cash flows.

Simple Interest (SI)

Topic: Time Value of MoneyChapter: Chapter 6
SI=P×R×T100SI = \frac{P \times R \times T}{100}
Calculates the simple interest earned or paid. P is the Principal amount, R is the annual interest rate (as a percentage, e.g., 10 for 10%), and T is the time period in years.

Amount (Future Value) with Simple Interest

Topic: Time Value of MoneyChapter: Chapter 6
A=P(1+R×T100)A = P \left(1 + \frac{R \times T}{100}\right)
Calculates the total amount (Principal + Simple Interest) at the end of the period. P is the Principal amount, R is the annual interest rate (as a percentage), and T is the time period in years.

Future Value (FV) with Compound Interest

Topic: Time Value of MoneyChapter: Chapter 6
FV=P(1+rm)m×t=P(1+i)nFV = P \left(1 + \frac{r}{m}\right)^{m \times t} = P(1+i)^n
Calculates the future value of a principal amount compounded periodically. P is the Principal (Present Value), r is the nominal annual interest rate (as a decimal), m is the number of compounding periods per year, t is the time in years, i is the interest rate per compounding period (r/m), and n is the total number of compounding periods (m*t).

Present Value (PV) with Compound Interest

Topic: Time Value of MoneyChapter: Chapter 6
PV=FV(1+rm)m×t=FV(1+i)−nPV = \frac{FV}{\left(1 + \frac{r}{m}\right)^{m \times t}} = FV(1+i)^{-n}
Calculates the present value of a future amount, discounted at a compound interest rate. FV is the Future Value, r is the nominal annual interest rate (as a decimal), m is the number of compounding periods per year, t is the time in years, i is the interest rate per compounding period (r/m), and n is the total number of compounding periods (m*t).

Effective Annual Rate (EAR)

Topic: Time Value of MoneyChapter: Chapter 6
EAR=(1+rm)m−1EAR = \left(1 + \frac{r}{m}\right)^m - 1
Calculates the actual annual rate of return or cost of funds when compounding occurs more frequently than once a year. r is the nominal annual interest rate (as a decimal), and m is the number of compounding periods per year.

Future Value of an Ordinary Annuity (FVOA)

Topic: Time Value of MoneyChapter: Chapter 6
FVOA=A[(1+i)n−1i]FVOA = A \left[\frac{(1+i)^n - 1}{i}\right]
Calculates the future value of a series of equal payments made at the end of each period. A is the periodic payment, i is the interest rate per period (as a decimal), and n is the total number of periods.

Future Value of an Annuity Due (FVAD)

Topic: Time Value of MoneyChapter: Chapter 6
FVAD=A[(1+i)n−1i](1+i)FVAD = A \left[\frac{(1+i)^n - 1}{i}\right] (1+i)
Calculates the future value of a series of equal payments made at the beginning of each period. A is the periodic payment, i is the interest rate per period (as a decimal), and n is the total number of periods.

Present Value of an Ordinary Annuity (PVOA)

Topic: Time Value of MoneyChapter: Chapter 6
PVOA=A[1−(1+i)−ni]PVOA = A \left[\frac{1 - (1+i)^{-n}}{i}\right]
Calculates the present value of a series of equal payments made at the end of each period. A is the periodic payment, i is the interest rate per period (as a decimal), and n is the total number of periods.

Present Value of an Annuity Due (PVAD)

Topic: Time Value of MoneyChapter: Chapter 6
PVAD=A[1−(1+i)−ni](1+i)PVAD = A \left[\frac{1 - (1+i)^{-n}}{i}\right] (1+i)
Calculates the present value of a series of equal payments made at the beginning of each period. A is the periodic payment, i is the interest rate per period (as a decimal), and n is the total number of periods.

Present Value of a Perpetuity

Topic: Time Value of MoneyChapter: Chapter 6
PVP=AiPVP = \frac{A}{i}
Calculates the present value of an infinite series of equal payments. A is the periodic payment, and i is the interest rate per period (as a decimal).
Chapter 6 Formulas — CA Foundation | Kartavvya