Question

Find the sum of sequence, 7, 77, 777, 7777..... to n terms ???​

Answer 1

= 7+77 +777+----+n

=7(1+11+111+----+n)

Multiplying and dividing 9

7/9 ( 9+99+999------+n)

=7/9 (10-1+100-1+1000-1------+n)

=7/9 ( 10-1+102-1+103-1-------+n)

Take all multiples of 10 together to form a GP and put all the -1's separate-

=7/9( 10+102+103....-n) (since there are 'n' terms so there are' n' no. of -1's so n times -1(n*-1) = -n)

Using formula for sum of GP

=7/9 { 10( 10n - 1)/9 - n}

= 70/81(10n-1) - n

Answer 2

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S= 7+77+777+7777+⋯⋯⋯n

S= 7(1+11+111+1111+⋯⋯⋯n

S= 7/9 (9+99+999+9999+⋯⋯⋯n)

S= 7/9 [ (10–1)+(100–1)+(1000–1)⋯⋯⋯n]

S= 7/9 [ (10+100+1000+⋯⋯⋯n) - (1+1+1+1⋯⋯⋯n)]

S= 7/9 [ (10+100+1000+⋯⋯⋯n) - n]

S= 7/9 [10(10ⁿ−1)/(10−1)−n]

S= 7/9 [10(10ⁿ−1/9)- n]  

S= 70/81[10ⁿ−1]−7n/9

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