The first two terms of an infinite GP ( geometric progression ) are together equal to 5 and every term is 3 times the sum of all the terms that follows it; find the series.

Question

terenzreignz · Answer

Your sequence should take the form of
$$\huge \{ ar^{n  }\}$$where a is your initial value and r is your common ratio.
Just find a and r.

terenzreignz · Answer

You know two things, the first two terms add up to 5, so...
$$a + ar = 5$$

You have two unknown, and one linear equation, maybe the second fact you know would lead to another one.  You need two linear equations to get to a definite answer if you have two unknowns.

anonymous · Answer

that  i have already done.. please let me know further

terenzreignz · Answer

Do you know how to evaluate an infinite geometric series (ie, the sum of the infinite terms of a geometric progression) ?

anonymous · Answer

$$S_{\infty} = \frac{a}{1-r}$$

terenzreignz · Answer

Where a is the initial term and r is the common ratio, correct.  Now it says that every term is equal to three times the sum of all the terms that follow it.  How do we interpret this?

anonymous · Answer

Sum of an infinite G.P is a multiple of $$\frac{r}{1-r}$$ factor  with a previous term that means : $$ar^n+ar^{n+1}+ar^{n+2}.....\infty $$ = $$ar^{n-1}\frac{r}{1-r}$$

terenzreignz · Answer

Simply put, for all non-negative integers, n

$$\huge ar^{n}=?$$

terenzreignz · Answer

Well, it says that every term is equal to the sum of the succeeding terms, ie, the rest of the geometric series, so...
$$\huge ar^{n}=\sum_{k=1}^{\infty}ar^{n+k}$$
Go ahead and simplify this.

shubhamsrg · Answer

or,,simply
1st term = 3* (sum of all the terms - 1st term)
4a = 3a/(1-r)

2 eqns, 2 variables..