Question

The sum of an infinite geometric sequence is 33.25.

The second term of the sequence is 7.98
Find the possible values of r

Answer 1

Possible values for r are 0.4 and 0.6.

Explanation:

The sum to infinity of a GS is a1 / (1 - r) where a1 is the first term and r is the common difference.

The second term = a1*r = 7.98

so a1 = 7.98 / r.

Substituting in the formula for the sum to infinity:

33 .25 = (7.98 / r) / (1 - r).

33.25 = 7.98 / r(1 - r)

33.25 = 7.98 / r - r^2

33.25r - 33.25r^2 = 7.98

33.25r^2 - 33.25r + 7.98 = 0

r = [ -(-33.25) +/- √(-33.25)^2 - 4 * 33.25 * 7.98)] / (2 * 33.25)

this gives r = 0.4, 0.6.

Check back in the formula for the infinite sum:

Let r = 0.4:

Then a1 =7.98 / 0.4 = 19.95

so sum = 19.95 / (1 - 0.4) = 33.25

Ler r = 0.6 then a1 = 7.98 / 0.6 = 13.3

and the sum = 13.3 / 1 - 0.6 = 33.35.

This confirms our values of r.

Answer 2

Let a be the first term.

The sum is a1−r=33.25.

The second term is ar=7.98, so a=7.98/r.

Putting these together, 7.98/r(1−r)=33.25 or r(1−r)=0.24=0.6×0.4.

If the answer doesn't jump out at you from there, you could solve for r with the quadratic formula.

Explanation:

I Hope It's Helpful :)