Question

solve the problem in the given image​

Answer 1

Step-by-step explanation:

→ 7 + 77 + 777 +... upto n terms

→ 7[ 1 + 11 + 111 +... upto n terms]

Divide and multiply by 9:

→ $\frac{7}{9}$[9 + 99 + 999 +... upto n terms]

→ $\frac{7}{9}$[(10 - 1) + (100 - 1) + (1000 - 1) +... upto n terms]

→ $\frac{7}{9}$[(10 + 100 + 1000+... upto n terms) + (-1 - 1 - 1 -... upto n terms)]

→ $\frac{7}{9}$[(10¹ + 10² + 10³ + upto n terms) - (1 + 1 + 1 +... upto n terms)]

10¹ + 10² + 10³... forms a GP, and sum till nth terms will be $\frac{10(10^n - 1)}{10-1}$, and

1 + 1 + 1... forms an AP, sum = $\frac{n}{2}(</strong><strong>1</strong><strong>+</strong><strong>1)$

Therefore,

→ $\frac{7}{9}[ \frac{10(10^n - 1)}{10-1} - \frac{n}{2}(1+1)]$

→ $\frac{7}{9}[ \frac{10(10^n - 1)}{9} -n]$

Similarly, for 6 + 66 + 666...

→ $\frac{6}{9}[ \frac{10(10^n - 1)}{9} -n]$

→ $\frac{2}{3}[ \frac{10(10^n - 1)}{9} - n]$