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.

Question

I understand how to sub in the formulas, but have been having trouble coming up with the right two possible answers, which are r=0.4 and r=0.6, and this will be on my upcoming assessment.

Answer 1

Important facts:

Let $a_n$ be a geometric series with multiplier $|r|<1$

• $\sum_{n=1}^\infty a_n = \frac{a_1}{1-r}$
• $a_2 = a_1 r \iff a_1 = \frac{a_2}{r}$

Now, it is a simple matter of solving $r$ for: $$\frac{a_1}{1-r} = 33.25$$

Lets write $a_1$ in terms of $a_2$: $$\frac{a_2}{r} \times \frac{1}{1-r} = \frac{7.98}{r(1-r)} = 33.25 $$

This boils down to solving the quadratic: $$r-r^2-\frac{7.98}{33.25} = 0 $$

Answer 2

Let $a$ be the first term.

The sum is $\dfrac a {1-r}=33.25.$

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

Putting these together, $\dfrac{7.98/r}{(1-r)}=33.25$ or $r(1-r)=0.24=0.6\times0.4$.

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