An infinite geometric progression of positive terms has a sum to infinity of 16, and the s… — JAMB UTME

An infinite geometric progression of positive terms has a sum to infinity of $16$ , and the sum of its first two terms is $12$ . An arithmetic progression has its first term equal to the first term of this geometric progression, and its $5^{\text{th}}$ term equal to the sum to infinity of the geometric progression. Calculate the $10^{\text{th}}$ term of the arithmetic progression.

Answer: 26

Answer

The 10th term of the arithmetic progression is 26.

By solving the geometric progression system, we find the common ratio

r = \frac{1}{2}

and first term

a = 8

. Using

a = 8

as the first term of the arithmetic progression and setting its

5^{\text{th}}

term

A_5 = 16

, we determine the common difference

d = 2

. Calculating

A_{10} = 8 + 9(2)

yields

26

Step-by-Step Solution

Formulate equations for the geometric progression using the sum to infinity and sum of the first two terms

a = 16(1 - r)

and

a(1 + r) = 12

The standard formula for the sum to infinity of a GP is

S_\infty = \frac{a}{1-r}

and the sum of the first two terms is

S_2 = a + ar = a(1+r)

Solve for the common ratio

r

and first term

a

of the geometric progression

r = 0.5

and

a = 8

Substituting

a = 16(1-r)

yields

16(1-r^2) = 12 \implies r^2 = \frac{1}{4} \implies r = \frac{1}{2}

. Then

a = 16(1 - 0.5) = 8

Determine the common difference

d

of the arithmetic progression

d = 2

The first term of the AP is

A_1 = a = 8

and the 5th term is

A_5 = S_\infty = 16

. Using

A_5 = A_1 + 4d \implies 8 + 4d = 16 \implies d = 2

Calculate the 10th term of the arithmetic progression

A_{10} = 26

Using the AP

n^{\text{th}}

term formula

A_n = A_1 + (n-1)d

A_{10} = 8 + 9(2) = 26

Key Concept

Combining geometric progression parameters (sum to infinity and sum of terms) with arithmetic progression term formulas

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