If the $n$-th term of a geometric progression $5,-\frac 52, \frac 54$.. is $\frac{5}{1024}$, then n is

Question

Common ratio = $r=\frac{-1}{2}$

First term=$a$

$$\frac{5}{1024}=5\left(-\frac{1}{2}\right)^{n-1}$$ $$\frac{1}{1024}=\left(-\frac{1}{2}\right)^{n-1}$$ $$1024=(-2)^{1-n}$$ $$(-2)^{10}=(-2)^{1-n}$$ Then $$10=1-n$$ $$n=-9$$ which makes so sense. How should I get the right answer?

The power should remain the same if you took the reciprocal of the base. i.e. it should be $$1024=(-2)^{n-1}$$ — Peter Foreman, Sep 15 '19 at 15:28

score 2 · Accepted Answer · edited Sep 15 '19 at 15:32

2

It started to be wrong from the third line. You take a $-1$ power both side but you do too much on your right side, it should be: $\left[\left(-\dfrac{1}{2}\right)^{n-1}\right]^{-1}=\left(-2\right)^{n-1}\text{ or }\left(-\dfrac{1}{2}\right)^{1-n}$

edited Sep 15 '19 at 15:32

Peter Foreman

19,947

answered Sep 15 '19 at 15:31

MafPrivate

4,033

I hope you don't mind me changing that. I thought using the $/$ was unclear and could be interpreted as division. – Peter Foreman Sep 15 '19 at 15:33
ok that's fine for me. It is better actually :D – MafPrivate Sep 15 '19 at 15:34
Why wouldn’t it change? Isn’t $\frac 1a = a^{-1}$? – Aditya Sep 15 '19 at 15:51
1

For $\left(a^b\right)^{-1}$, it can be expressed as $a^{-b}$ or $\left(\dfrac{1}{a}\right)^b$, but not $\left(\dfrac{1}{a}\right)^{-b}$ – MafPrivate Sep 15 '19 at 16:26

Quanto · Answer 2 · 2019-09-15T15:36:05.457

1

Use

$$a_n = a_1 r^{(n-1)}$$

Plug in $r=-1/2$, $a_1=5$ and $a_n =5/1024$ to get

$$n=11$$

edited Sep 15 '19 at 15:36

answered Sep 15 '19 at 15:33

Quanto

97,352

The OP clearly applies this formula but makes a mistake. – Peter Foreman Sep 15 '19 at 15:35
Actually, he knows the sequence, but he has something doing wrong in while doing this question, so please answer correctly – MafPrivate Sep 15 '19 at 15:35
Sure, I mean I could make out that it was 11, but I needed a proper way to solve it. – Aditya Sep 15 '19 at 15:50
@Aditya Sorry for misunderstanding – Quanto Sep 15 '19 at 16:05

If the $n$-th term of a geometric progression $5,-\frac 52, \frac 54$.. is $\frac{5}{1024}$, then n is

2 Answers2