SOLUTION: the fourth term of a GP is 1/2 and the sixth term is 1/8. its common ratio is negative a) how many term of GP add up to give a sum of -2/22/32? b)what is the sum to infinity for

Question 1116337: the fourth term of a GP is 1/2 and the sixth term is 1/8. its common ratio is negative
a) how many term of GP add up to give a sum of -2/22/32?
b)what is the sum to infinity for this GP
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
ONE WAY TO START:
The recursive relation for a geometric progression,
relating term number , to the preceding term, is
, where is the common ratio.
So, applying that to the fourth, fifth and sixth terms
,
, and
--> --> .
Knowing that , we conclude that ,
and can write the first 4 terms, starting from the 4th,
and dividing by (multiplying times -2)
each time to get the term before:
,
,
,
.

ANOTHER WAY:
The general "formula" for term number , ,
of a GP with first term and common ratio is
.
Applying that to ,
--> --> --> -->

AND FROM THERE:
The "formula" for the sum of the first terms, , of a GP is

When ,
the term tends to as tends to ,
and the sum to infinity is
.
In this case the sum to infinity is easy to calculate:

Substituting the and values found above,
the expression for the sum of the first terms becomes

If that is supposed to be an irreducible fraction of the form ,
the only possibility is ,
and as , --> .

BOTH QUESTIONS WERE ANSWERED, BUT IF YOU WANT MORE,
knowing , we could try to find that denominator.
.
So,

.
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