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Series , 7 , 77, 777 , 7777, .....

we use the geometric series to be able to do this.

Tn = 7 * 11111 ... n times

= 7 * [ 1 + 10 + 100 + .... + 10^{n-1} ]

= 7 * [ 10^n - 1] / [ 10 - 1 ]

= 7 * [10^n - 1 ] / 9

Sum of first n terms Tn is =

Sn = Σ (7/9) * 10^n - Σ (7/9)

= (7/9) * Σ 10^n - 7 n /9

= 7/9 * 10 * (10^n -1) / (10 - 1) - 7 n / 9

= 70 (10^n -1 ) / 81 - 7 n / 9

= [ 70 * 10^n - 63 n - 70 ] / 81

we use the geometric series to be able to do this.

Tn = 7 * 11111 ... n times

= 7 * [ 1 + 10 + 100 + .... + 10^{n-1} ]

= 7 * [ 10^n - 1] / [ 10 - 1 ]

= 7 * [10^n - 1 ] / 9

Sum of first n terms Tn is =

Sn = Σ (7/9) * 10^n - Σ (7/9)

= (7/9) * Σ 10^n - 7 n /9

= 7/9 * 10 * (10^n -1) / (10 - 1) - 7 n / 9

= 70 (10^n -1 ) / 81 - 7 n / 9

= [ 70 * 10^n - 63 n - 70 ] / 81