Question

The sum of an infinite g. P. With positive terms is 48 and sum of its first two terms is 36. Find the second term

Answer 1

The second term is 12.

Step-by-step explanation:

The geometric progression is in the form $a,ar,ar^2,ar^3,....$

Where, a is the first term and r is the common ratio.

The infinite sum of an G.P. is $S_n=\frac{a}{1-r}$

We have given the sum of an infinite G.P is 48.

i.e. $48=\frac{a}{1-r}$

$a=48(1-r)$ .....(1)

The sum of its first two terms is 36.

i.e. $a+ar=36$

$a(1+r)=36$

$a=\frac{36}{1+r}$ .....(2)

Equate (1) and (2),

$48(1-r)=\frac{36}{1+r}$

$(1-r)(1+r)=\frac{36}{48}$

$1-r^2=\frac{3}{4}$

$r^2=1-\frac{3}{4}$

$r^2=\frac{1}{4}$

$r=\pm\frac{1}{2}$

Substitute r in (1),

When $r=\frac{1}{2}$

$a=48(1-\frac{1}{2})$

$a=48(\frac{1}{2})$

$a=24$

The second term is $a_2=ar$

$a_2=24\times \frac{1}{2}$

$a_2=12$

When $r=-\frac{1}{2}$

$a=48(1-(-\frac{1}{2}))$

$a=48(\frac{3}{2})$

$a=72$

The second term is $a_2=ar$

$a_2=72\times -\frac{1}{2}$

$a_2=-36$

We reject because G.P is with positive terms.

Therefore, the second term is 12.

#Learn more

The sum of first two terms of an infinite gp is 1 and every term is twice the sum of successive terms. find its first term .

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