Find the sum of following sequence up to n terms 7, 77, 777, 7777

1.	Find the sum of following sequence up to n terms 7, 77, 777, 7777
Answer» Let, S = 7 + 77 + 777 + …77 … 7 (upto n terms) S = 7[1+ 11 + 111 + …11 … 1] (upto n terms) s = \(\frac{7}{9}\)[9 + 99 + 999 + ⋯ 99 … 9] s = \(\frac{7}{9}\)[10 − 1 + 10² − 1 + 10³ − 1 + ⋯ 10ⁿ − 1] s = \(\frac{7}{9}\)[10¹ + 10² + ⋯ 10ⁿ − n] s = \(\frac{7}{9}\)\(\big[\frac{10(10^n − 1)}{10 − 1} − n\big]\) s = \(\frac{7}{9}\)\(\big[\frac{10}{9}(10^n-1) − n\big]\)

Answer»

Let, S = 7 + 77 + 777 + …77 … 7 (upto n terms)

S = 7[1+ 11 + 111 + …11 … 1] (upto n terms)

s = \(\frac{7}{9}\)[9 + 99 + 999 + ⋯ 99 … 9]

s = \(\frac{7}{9}\)[10 − 1 + 10² − 1 + 10³ − 1 + ⋯ 10ⁿ − 1]

s = \(\frac{7}{9}\)[10¹ + 10² + ⋯ 10ⁿ − n]

s = \(\frac{7}{9}\)\(\big[\frac{10(10^n − 1)}{10 − 1} − n\big]\)

s = \(\frac{7}{9}\)\(\big[\frac{10}{9}(10^n-1) − n\big]\)

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