Question

The first term of an infinite G. P.is 4 and every term is equal to 3 times the sum of all the following terms.The common ratio is ​

Answer 1

The common ratio of the infinite G.P. is equal to - 2

Given that;

The first term of an infinite G. P. is 4 and every term is equal to 3 times the sum of all the following terms.

To find;

The common ratio

Solution;

Let a= first term & r= common ratio of G.P.

So, according to the question we have,

a = 4. Also, we know that the infinite series of G.P. is given by,

a + a$r^{}$ + a$r^{2}$ + $ar^{3}$ + ...

Now, the sum of an infinite series of G.P. is given by,

S = $\frac{a}{1 - r}$

According to the question we have,

Every term is equal to 3 times the sum of all the following terms i.e.

3( a$r^{}$ + a$r^{2}$ + $ar^{3}$ + ... )= 4 (first term) therefore,

3 x $\frac{a}{1 - r}$ = 4. putting the value of a = 4 i.e. first term in the given sum formula we get,

3 x $\frac{4}{1 - r}$ = 4

3 x 4 = 4(1 - r)

3 = 1 -r

r = - 2

Hence, The common ratio = - 2

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