Question

Find the sum of series: 7 + 77 + 777 + …………………. To n terms.

Answer 1

Answer

• 7/9 [ 10 × ( 10ⁿ - 1 ) / 9] - n

Given

• 7 + 77 + 777 + ..........

To Find

• Sum of series

Solution

$\tt 7+77+777+.....n\\\\\implies \tt 7(1+11+111+.....n/7)\\\\\implies \sf 7(1+11+111+.....n/7)\times \dfrac{9}{9}\;\;\; [Multiply\ with\ \dfrac{9}{9} ]\\\\\implies \tt \dfrac{7}{9}(9+99+999+.....+9n/7)\\\\\implies \tt \dfrac{7}{9}([10-1]+[100-1]+[1000-1]+.....+[10^n-1])\\\\\implies \tt \dfrac{7}{9}\bigg(10+10*10+10*10^2+..... \bigg) -1*n$

Now use Sum of n terms in Geometric Progression .$\tt S_n=\dfrac{a(r^n-1)}{r-1}$

$\implies \tt \dfrac{7}{9}\bigg( \dfrac{10(10^n-1)}{10-1} \bigg)-n\\\\\implies \tt \dfrac{7}{9}\bigg( \dfrac{10(10^n-1)}{9} \bigg)-n$