Question

SECOND TERM AND FIFTH TERMS OF GEOMETRIC SERIES ARE -1/2 AND 1/16 RESPECTIVELY FIND SUM OF THE SERIES UP TO 8 TERMS

Answer 1

I'm still in class Xth, so mistakes are possible.......... : p

Answer 2

Answer:

85/128

Step-by-step explanation:

The nth term of a GP can be given as: arⁿ⁻¹

It is given that Second term is  -1/2

⇒ ar²⁻¹ = -1/2

⇒ ar = -1/2  ... ( 1 )

Also it is given that, Fifth term is 1/16

⇒ ar⁵⁻¹ = 1/16

⇒ ar⁴ = 1/16  ... ( 2 )

Dividing ( 2 ) by ( 1 ) we get,

$\implies \dfrac{ ar^4}{ar} = \dfrac{ \dfrac{1}{16} }{ \dfrac{-1}{2} }\\\\\text{ This can also be written as} \\\\\implies r^3 =\dfrac{1}{16} \times \dfrac{-2}{1} \implies \dfrac{-2}{16} \implies \dfrac{-1}{8}\\\\\text{ Taking Cube root on both sides we get,}\\\\\implies r =\sqrt[3]{ \dfrac{-1}{8}} \implies \dfrac{-1}{2}$

So common ratio is -1/2.

Substituting this in ( 1 ) we get,

⇒ ar = -1/2

⇒ a ( -1/2 ) = -1/2

⇒ a = 1

So First term is 1.

Sum of terms in a GP is given by:

$\text{ Sum } = \dfrac{ a( 1 - r^n )}{ 1 - r } \:\:\text{Since r is less than 1 }\\\\\text{ Substituting the values we get,}\\\\\implies Sum = \dfrac{ 1 ( 1 - \dfrac{-1}{2}^8)}{1 - \dfrac{-1}{2} }\\\\\implies \dfrac{-1^8}{2^8} = \dfrac{1}{256} \\\\\implies Sum = \dfrac{ 1 -\dfrac{ 1}{256} }{ 1.5}\\\\\implies Sum = \dfrac{255}{256} \times \dfrac{2}{3} \implies\dfrac{85}{128}$

Hence the sum of the firsts 8 terms is 85/128.