Find the value of 7 + 77 + 777 + 777 + …………..+ n terms

Solution

7 + 77 + 777 + 777 +…………..+ n terms 7(1 + 11 + 111 + ………………………+ n terms) Multiply and divide by 9 9/9 × 7(1 + 11 + 111 +………………………+ n terms) 7/9 × (9 + 99 + 999 + …………………+ n terms) 7/9 × [(10 - 1) + (100 - 1) + (1000 - 1) +…………… + n terms] 7/9 × [(10 + 10² + 10³ +………+ n terms) – (1+1+1+………+ n terms)] 7/9 × [(10 + 10² + 10³ +………+ n terms) – n] In the series [10 + 10² + 10³ +………+ n terms] First term, a = 10 common ratio, r = 10 Sum of n terms = (a(rⁿ-1))/((r-1)) ∴ Required answer = 7/9 [((10(10ⁿ-1))/9 - n)]