Question

solve this ..........​

Answer 1

$♕ \: \: \large{\rm{{\underline{\underline{\red{S}\purple{O}\pink{LU}\orange{TI}\green{ON}}}}}} \: \: ♕$

FORMULA TO BE IMPLEMENTED

For an Geometric Progression with first term = a and common ratio = r ( < 1)

The sum of first n terms

$\displaystyle \: = a \: \times \frac{1 - {r}^{n} }{1 - r}$

TO DETERMINE

$7 + 7.7 + 7.77 + 7.777 + ............upto \: \: 50 \: \: terms$

CALCULATION

$7 + 7.7 + 7.77 + 7.777 + ............upto \: \: 50 \: \: terms$

$= \displaystyle \: \: 7 \times (1 + 1.1 + 1.11 + 1.111 + ............upto \: \: 50 \: \: terms$

$= \displaystyle \: \: \dfrac{7}{9} \times (9 + 9.9 + 9.99 + 9.999 + ............upto \: \: 50 \: \: terms$

$= \displaystyle \: \: \dfrac{7}{9} \times \{ \: (10 - 1) + (10 - 0.1) +(10 - 0.01)+ (10 - 0.001)+ ............upto \: \: 50 \: \: terms \}$

$= \displaystyle \: \: \dfrac{7}{9} \times \{ \: (10 + 10 + 10 + 10 + ........upto \: \: 50 \: \: terms \: ) - (1 + 0.1 + 0.01 + 0.001 + ............upto \: \: 50 \: \: terms \}$

$= \displaystyle \: \: \dfrac{7}{9} \times \{ \: (10 \times 50 ) - (1 + 0.1 + 0.01 + 0.001 + ............upto \: \: 50 \: \: terms \}$

$= \displaystyle \: \: ( \dfrac{7}{9} \times \ \: 500 ) - \frac{7}{9} (1 + 0.1 + 0.01 + 0.001 + ............upto \: \: 50 \: \: terms \} \: \: .........(1)$

Now

$= \displaystyle \: 1 + 0.1 + 0.01 + 0.001 + ............upto \: \: 50 \: \: terms$

This is a Geometric Progression

Where First term = a = 1

Common Ratio = r = 0.1 < 1

So

$\displaystyle \: 1 + 0.1 + 0.01 + 0.001 + ............upto \: \: 50 \: \: terms$

$= \displaystyle \: 1 \times \frac{1 - {(0.1)}^{50} }{1 - 0.1}$

$\displaystyle \: = \frac{10}{9} \{1 - {(0.1)}^{50} \}$

Hence from (1)

$7 + 7.7 + 7.77 + 7.777 + ............upto \: \: 50 \: \: terms$

$\displaystyle \: = \frac{3500}{9} - \frac{7}{9} \times \frac{10}{9} \{1 - {(0.1)}^{50} \}$

$\displaystyle \: = \frac{3500}{9} - \frac{70}{81} \{1 - {(0.1)}^{50} \}$