Question

find the sum of first 100 terms of the sequence 0.7,0.07,0.007,........... ?​

Answer 1

$\LARGE{ \underline{ \red{ \sf{Required \: answer:}}}}$

The terms in the given sequence are related in some ratio, which means it so said to be in geometric progression. The constant is known as common ratio.

Here,

• a = 0.7
• r = 0.07 / 0.7 = 0.1
• n = 100

Formula for sum of n terms in a GP:

$\large{ \because{ \boxed{ \rm{S_n = \frac{a(1 - {r}^{n}) }{1 - r} }}}}$

Because r is less than 1 here.

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Plugging the given values in the formula,

⇛ $\sf{S_{100} = \dfrac{0.7(1 - {0.1}^{100}) }{1 - 0.1} }$

Solving further,

⇛ $\sf{S_{100} = \dfrac{0.7(1 - {0.1}^{100}) }{0.9} }$

Now this can be written as,

⇛ $\sf{S_{100} = \dfrac{7}{9} \times \dfrac{9999....99(100 \: times)}{ {10}^{100} } }$

Simplifying further,

⇛ $\sf{S_{100} = \dfrac{777....77(100 \: times)}{ {10}^{100} }}$

Sum of 100 terms of the GP:

⇛ $\sf{S_{100} = 0.7777...77(100\: times)}$

And we are done !!

Answer 2

${ \huge\underline{\red{\sf{Required \: Answer}}}}$

The terms in the given sequence are related in some ratio, which means it so said to be in geometric progression. The constant is known as common ratio.

Here,

• a=0.7
• r=0.07/0.7=0.1
• n=100

Formula for sum of n terms in a GP:

$\large{ \because{ \boxed{ \rm{S_n = \frac{a(1 - {r}^{n}) }{1 - r} }}}}$

Because r is less than 1 here.

━━━━━━━━━━━━━━━━━━━━

Plugging the given values in the formula,

$\\ ⇛ \sf{S_{100} = \dfrac{0.7(1 - {0.1}^{100}) }{1 - 0.1} }$

Solving further,

S=0.777.......7777(100 times)