Geometric Sequence with formula for kth term.

Question

The problem looks like this

$\text{Sequence}$

$$-7\frac{1}{2},2 \frac{13}{16}, -1\frac{7}{128},\frac{405}{1024},-\frac{1215}{8192}$$

Thanks to the final two numbers, I was able to recognize the ratio$\frac38$. I know this is a geometric sequence, so the ratio would be $\frac38 ^k$. Given that it oscillates back and forth and it starts on a negative number, I would think that my formula would include a $-1^{k+1}$ for $n \ge 0$.

Since I have my original value $a = -7\frac12$ and my ration $\frac38^k$ as well as my oscillator $-1^{k+1}$, I would think that my function would be

$$a_k = 7\frac12 \cdot \left(\frac38\right)^k (-1)^{k+1}.$$

Does this look correct?

Answer 1

If you mean $a_k = 7\frac12 \left(\frac38\right)^k (-1)^{k+1}$ then your formula is wrong for $k=1$.

It should be $$a_{k}=-7\frac{1}{2}\left(-\frac{3}{8}\right)^{k-1}.$$

If you want that $a_0=-7\frac{1}{2}$ then it should be $$a_{k}=-7\frac{1}{2}\left(-\frac{3}{8}\right)^{k}.$$

Answer 2

The ratios of successive terms (second over first) are

$$-\frac38,-\frac38,-\frac38,-\frac38$$

and you indeed have a geometric progression of common ratio $-\dfrac38$.

Hence $$a_k=a_0r^k=-7\frac12\left(-\frac38\right)^k.$$