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

sublimnoj7u7

sublimnoj7u7

Answered question

2023-01-11

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

Answer & Explanation

Keely Franco

Keely Franco

Beginner2023-01-12Added 11 answers

Given : sequence 7 , 77 , 777 , 7777 , . . . upto n terms
Here, 77 7 = 11
and 777 77 = 10.09
∵ The common ratio differs.
∴ The given sequence is not G.P
We must identify the sum = 7 + 77 + 777 + 7777 + upto n terms
= 7 ( 1 + 11 + 111 + upto n terms )
Multiplying & dividing by 9
= 7 9 [ 9 ( 1 + 11 + 111 + . . .  upto n terms ) ]
= 7 9 [ 9 + 99 + 999 + 9999 + . . .  upto n terms ]
= 7 9 [ ( 10 1 ) + ( 100 1 ) + ( 1000 1 ) + . . . upto n terms ]
= 7 9 [ ( 10 + 100 + 1000 + . . . n terms ) ( 1 + 1 + 1 + . . . upto n terms ]
Sum = 7 9 [ ( 10 + 100 + 1000 + . . .  terms ) n × 1 ]⋯(i)
Now, 10 + 100 + 1000 + . . . n terms is a G.P
Here, a=10 and r=10>1
We know,
sum of n terms of G.P., = S n = a ( r n 1 ) r 1 ; r > 1
S n = 10 ( 10 n 1 ) 10 1
S n = 10 ( 10 n 1 ) 9
Now substituting this value in (i), we get
Sum = 7 9 [ 10 ( 10 n 1 ) 9 n ]

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