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Worked solution

Recover a geometric ratio from a term and infinite sum

The second term of a geometric progression is 55 while its sum to infinity is 2020. Find the sum of the first 88 terms.

CJC · Sequences and Series · Example 27

Answer

S8=127564S_8=\frac{1275}{64}.

Compare your reasoning

Translate the two given quantities

u2=ar=5(1),S∞=a1−r=20(2)u_2=ar=5\quad\text{(1)},\qquad S_\infty=\frac{a}{1-r}=20\quad\text{(2)}

A sum to infinity is given, so the nonzero GP has |r|<1. Retain the two source equations.

Divide the equations to eliminate a

r(1−r)=14,r2−r+14=0,(r−12)2=0,r=12.\begin{aligned}r(1-r)&=\frac14,\\r^2-r+\frac14&=0,\\\left(r-\frac12\right)^2&=0,\\r&=\frac12.\end{aligned}

Equation (1) divided by equation (2) leaves r(1−r). Complete the square as in the source.

Recover the first term

a=5(2)=10a=5(2)=10

The source continuation on page 42 substitutes in equation (1).

Evaluate the finite sum requested

S8=10[1−(1/2)8]1−1/2=127564\begin{aligned}&S_8\\ & =\frac{10[1-(1/2)^8]}{1-1/2}\\ & =\frac{1275}{64}\end{aligned}

Use eight terms, rather than reporting the given infinite sum.

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