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Chapter 9

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Arithmetic Progression (AP) Term

Topic: Logical ReasoningChapter: Chapter 9
an=a1+(n−1)da_n = a_1 + (n-1)d
This formula defines the nth term of an Arithmetic Progression. Each term after the first (a_1) is obtained by adding a constant difference (d) to the preceding term. 'n' represents the term number. This pattern is fundamental in many number series problems.

Geometric Progression (GP) Term

Topic: Logical ReasoningChapter: Chapter 9
an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}
This formula defines the nth term of a Geometric Progression. Each term after the first (a_1) is obtained by multiplying the preceding term by a constant ratio (r). 'n' represents the term number. This pattern is also common in number series where terms grow or shrink exponentially.

Difference Series Rule

Topic: Logical ReasoningChapter: Chapter 9
di=ai+1−aid_i = a_{i+1} - a_i
This rule describes a series where the differences between consecutive terms (d_i) themselves form a new series. This 'difference series' might follow an AP, GP, or another recognizable pattern. This method can be extended to 'double difference' or 'triple difference' series to find the underlying pattern.

Square/Cube Series Rule

Topic: Logical ReasoningChapter: Chapter 9
Tn=n2 or n3 or n2±k or n3±kT_n = n^2 \text{ or } n^3 \text{ or } n^2 \pm k \text{ or } n^3 \pm k
This rule indicates that terms in a series are derived from squares or cubes of natural numbers (n), often with a constant addition or subtraction (k). Recognizing these fundamental number properties is key to solving many complex series.

Alphabetical Shift (Coding-Decoding)

Topic: Logical ReasoningChapter: Chapter 9
Pcoded=(Poriginal+k)(mod26)P_{coded} = (P_{original} + k) \pmod{26}
In letter coding, this rule represents a 'shift cipher' where each letter's numerical position (P) in the alphabet is shifted by a constant value (k). The modulo 26 ensures that the alphabet wraps around (e.g., if Z is shifted by 1, it becomes A). 'k' can be positive (forward shift) or negative (backward shift).

Alphabetical Position (Coding-Decoding)

Topic: Logical ReasoningChapter: Chapter 9
A=1,B=2,...,Z=26 or Z=1,Y=2,...,A=26A=1, B=2, ..., Z=26 \text{ or } Z=1, Y=2, ..., A=26
This rule assigns numerical values to letters based on their sequential position in the alphabet, either in forward or reverse order. These numerical values are then used in mathematical operations (addition, subtraction, multiplication) to form codes or decode messages.

Simple Interest (SI)

Topic: Time Value of MoneyChapter: Chapter 9
SI=P×R×T100SI = \frac{P \times R \times T}{100}
This formula calculates the interest earned only on the principal amount over a given period. P is the Principal amount, R is the Rate of Interest per annum, and T is the Time period in years. The total amount (A) would be P + SI.

Compound Interest (CI) - Future Value (FV)

Topic: Time Value of MoneyChapter: Chapter 9
FV=P(1+Rn)n×TFV = P \left(1 + \frac{R}{n}\right)^{n \times T}
This formula calculates the future value of an investment where interest is compounded periodically. P is the Principal amount, R is the annual nominal interest rate, T is the time in years, and n is the number of compounding periods per year. The compound interest (CI) itself is FV - P.

Compound Interest (CI) - Present Value (PV)

Topic: Time Value of MoneyChapter: Chapter 9
PV=FV(1+Rn)n×TPV = \frac{FV}{\left(1 + \frac{R}{n}\right)^{n \times T}}
This formula calculates the present value of a future sum of money, discounted back to the present. FV is the Future Value, R is the annual nominal interest rate, T is the time in years, and n is the number of compounding periods per year.

Future Value of an Ordinary Annuity (FVA)

Topic: Time Value of MoneyChapter: Chapter 9
FVA=Pmt×[(1+i)n−1i]FVA = Pmt \times \left[\frac{\left(1 + i\right)^n - 1}{i}\right]
This formula calculates the future value of a series of equal payments (Pmt) made at the end of each period. 'i' is the interest rate per period, and 'n' is the total number of periods. An ordinary annuity assumes payments occur at the end of each period.

Present Value of an Ordinary Annuity (PVA)

Topic: Time Value of MoneyChapter: Chapter 9
PVA=Pmt×[1−(1+i)−ni]PVA = Pmt \times \left[\frac{1 - \left(1 + i\right)^{-n}}{i}\right]
This formula calculates the present value of a series of equal payments (Pmt) made at the end of each period. 'i' is the interest rate per period, and 'n' is the total number of periods. This is used to find out how much a stream of future payments is worth today.

Future Value of an Annuity Due (FVAD)

Topic: Time Value of MoneyChapter: Chapter 9
FVAD=Pmt×[(1+i)n−1i]×(1+i)FVAD = Pmt \times \left[\frac{\left(1 + i\right)^n - 1}{i}\right] \times \left(1 + i\right)
This formula calculates the future value of a series of equal payments (Pmt) made at the beginning of each period. 'i' is the interest rate per period, and 'n' is the total number of periods. Annuity due payments earn one extra period of interest compared to an ordinary annuity.

Present Value of an Annuity Due (PVAD)

Topic: Time Value of MoneyChapter: Chapter 9
PVAD=Pmt×[1−(1+i)−ni]×(1+i)PVAD = Pmt \times \left[\frac{1 - \left(1 + i\right)^{-n}}{i}\right] \times \left(1 + i\right)
This formula calculates the present value of a series of equal payments (Pmt) made at the beginning of each period. 'i' is the interest rate per period, and 'n' is the total number of periods. This is used when payments are made at the start of each period.

Present Value of a Perpetuity

Topic: Time Value of MoneyChapter: Chapter 9
PVperpetuity=PmtiPV_{perpetuity} = \frac{Pmt}{i}
This formula calculates the present value of an infinite series of equal payments (Pmt) that continues indefinitely. 'i' is the interest rate per period. This assumes the first payment occurs at the end of the first period.

Effective Annual Rate (EAR)

Topic: Time Value of MoneyChapter: Chapter 9
EAR=(1+Rn)n−1EAR = \left(1 + \frac{R}{n}\right)^n - 1
This formula calculates the actual annual rate of interest earned or paid on an investment or loan, considering the effect of compounding. R is the nominal annual interest rate, and n is the number of compounding periods per year.

Simple Interest (Amount)

Topic: Time Value of MoneyChapter: Chapter 9
A=P(1+RT)A = P(1 + RT)
Calculates the total amount (Principal + Interest) after a period, where interest is earned only on the initial principal. A is the total amount, P is the principal, R is the annual interest rate (as a decimal), and T is the time period in years.

Compound Interest (Future Value of a Single Sum)

Topic: Time Value of MoneyChapter: Chapter 9
FV=P(1+i)nFV = P(1 + i)^n
Calculates the future value of a single sum invested today, compounded over time. FV is the future value, P is the principal (present value), i is the interest rate per compounding period, and n is the total number of compounding periods.

Compound Interest (Present Value of a Single Sum)

Topic: Time Value of MoneyChapter: Chapter 9
PV=FV(1+i)−nPV = FV(1 + i)^{-n}
Calculates the present value of a single sum to be received in the future. PV is the present value, FV is the future value, i is the interest rate per compounding period, and n is the total number of compounding periods.

Future Value of an Ordinary Annuity

Topic: Time Value of MoneyChapter: Chapter 9
FVOA=PMT[(1+i)n−1i]FVOA = PMT \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. FVOA is the future value of the ordinary annuity, PMT is the payment per period, i is the interest rate per period, and n is the number of payments.

Future Value of an Annuity Due

Topic: Time Value of MoneyChapter: Chapter 9
FVAD=PMT[(1+i)n−1i](1+i)FVAD = PMT \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. FVAD is the future value of the annuity due, PMT is the payment per period, i is the interest rate per period, and n is the number of payments.

Present Value of an Ordinary Annuity

Topic: Time Value of MoneyChapter: Chapter 9
PVOA=PMT[1−(1+i)−ni]PVOA = PMT \left[ \frac{1 - (1 + i)^{-n}}{i} \right]
Calculates the present value of a series of equal payments to be received at the end of each period. PVOA is the present value of the ordinary annuity, PMT is the payment per period, i is the interest rate per period, and n is the number of payments.

Present Value of an Annuity Due

Topic: Time Value of MoneyChapter: Chapter 9
PVAD=PMT[1−(1+i)−ni](1+i)PVAD = PMT \left[ \frac{1 - (1 + i)^{-n}}{i} \right] (1 + i)
Calculates the present value of a series of equal payments to be received at the beginning of each period. PVAD is the present value of the annuity due, PMT is the payment per period, i is the interest rate per period, and n is the number of payments.

Effective Annual Interest Rate

Topic: Time Value of MoneyChapter: Chapter 9
EIR=(1+rm)m−1EIR = \left(1 + \frac{r}{m}\right)^m - 1
Calculates the actual annual rate of interest earned or paid, considering the effect of compounding more frequently than once a year. EIR is the effective annual interest rate, r is the nominal annual interest rate (as a decimal), and m is the number of compounding periods per year.

Arithmetic Progression (AP) - nth Term

Topic: Logical Reasoning PatternsChapter: Chapter 9
an=a1+(n−1)da_n = a_1 + (n-1)d
This formula calculates the nth term (a_n) of an arithmetic progression. Here, 'a_1' is the first term, 'n' is the term number, and 'd' is the common difference between consecutive terms. Use this when you observe a constant difference between numbers in a series.

Geometric Progression (GP) - nth Term

Topic: Logical Reasoning PatternsChapter: Chapter 9
an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}
This formula calculates the nth term (a_n) of a geometric progression. Here, 'a_1' is the first term, 'n' is the term number, and 'r' is the common ratio by which each term is multiplied to get the next. Use this when you observe a constant ratio between numbers in a series.

Fibonacci Series

Topic: Logical Reasoning PatternsChapter: Chapter 9
Fn=Fn−1+Fn−2F_n = F_{n-1} + F_{n-2}
This recursive formula defines the Fibonacci series, where each number is the sum of the two preceding ones. The series typically starts with F_0 = 0 and F_1 = 1 (or F_1 = 1 and F_2 = 1). Look for this pattern when numbers grow rapidly and each term is a sum of the previous two.

Square Series

Topic: Logical Reasoning PatternsChapter: Chapter 9
an=n2 or n2±ka_n = n^2 \text{ or } n^2 \pm k
This represents a series where terms are squares of natural numbers (n), or squares adjusted by a constant 'k'. Identify this pattern by recognizing perfect squares or numbers close to perfect squares.

Cube Series

Topic: Logical Reasoning PatternsChapter: Chapter 9
an=n3 or n3±ka_n = n^3 \text{ or } n^3 \pm k
This represents a series where terms are cubes of natural numbers (n), or cubes adjusted by a constant 'k'. Identify this pattern by recognizing perfect cubes or numbers close to perfect cubes.

Difference Series

Topic: Logical Reasoning PatternsChapter: Chapter 9
dn=an+1−and_n = a_{n+1} - a_n
This is not a direct formula for the term itself, but a method to find the pattern. Calculate the differences between consecutive terms (d_n). If the first differences don't show a pattern, calculate the differences of the differences (second differences), and so on. The pattern might emerge in these successive differences.

Coding - Alphabet Positional Value

Topic: Logical Reasoning PatternsChapter: Chapter 9
PL=position of letter L in standard alphabet (A=1, B=2, ..., Z=26)P_L = \text{position of letter L in standard alphabet (A=1, B=2, ..., Z=26)}
This rule assigns a numerical value to each letter based on its position in the English alphabet. Many coding-decoding problems rely on these values, either directly or in reverse (Z=1, Y=2, ...). Use this to convert letters to numbers for arithmetic operations.

Coding - Shifting/Substitution Rule

Topic: Logical Reasoning PatternsChapter: Chapter 9
CL=L±k (for letter position) or CL=Substitute(L)C_L = L \pm k \text{ (for letter position) or } C_L = \text{Substitute(L)}
This represents rules for letter coding. 'L' is the original letter's position or identity. 'k' is a constant shift (e.g., Caesar cipher: each letter shifts by 3 positions). Substitution involves replacing a letter with another based on a specific rule (e.g., opposite letters, specific mapping). The 'formula' here describes the operation applied to each letter.

Nth Term of an Arithmetic Progression (AP)

Topic: Logical Reasoning PatternsChapter: Chapter 9
an=a1+(n−1)da_n = a_1 + (n-1)d
Calculates the nth term (ana_n) of an arithmetic progression, where a1a_1 is the first term, nn is the term number, and dd is the common difference between consecutive terms.

Nth Term of a Geometric Progression (GP)

Topic: Logical Reasoning PatternsChapter: Chapter 9
an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}
Calculates the nth term (ana_n) of a geometric progression, where a1a_1 is the first term, nn is the term number, and rr is the common ratio between consecutive terms.

Sum of First N Natural Numbers

Topic: Logical Reasoning PatternsChapter: Chapter 9
Sn=n(n+1)2S_n = \frac{n(n+1)}{2}
Calculates the sum (SnS_n) of the first nn positive integers (1, 2, 3, ..., n). This pattern often appears in the differences of terms in a series or as the terms themselves.
Chapter 9 Formulas — CA Foundation | Kartavvya