Question

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Answer 1

Answer:(7/9) [ (10/9) ( 10ⁿ - 1 ) - n ]...................

Step-by-step explanation:7+77+777+....... upto n terms.

7(1+11+111+........upto n terms)

7/9(9+99+999+.........upto n terms)

7/9[(10-1)+(10²-1)+...... upto n terms]

7/9[10+10²+.........+10ⁿ}-(1+1+1+....n times)]

(7/9) { 10 [ ( 10ⁿ - 1 ) / ( 10 - 1 ) ] - n(1) }

(7/9) [ (10/9) ( 10ⁿ - 1 ) - n ]...................

Answer 2

Answer:

Step-by-step explanation:

( 7 + 77 + 777 + 7777 + .... n terms )

Taking 7 in common we get,

⇒ 7 ( 1 + 11 + 111 + 1111 + .... n terms )

Multiplying and dividing by 9 we get,

⇒ 7/9 ( 9 + 99 + 999 + 9999 + ... n terms )

⇒ 7/9 ( [ 10 - 1 ] + [ 100 - 1 ] + [ 1000 - 1 ] + .... n terms )

⇒ 7/9 ( 10 + 100 + 1000 + ..... n terms - 1 - 1 - 1 .... n terms )

⇒ 7/9 ( 10 + 100 + 1000 + ... n terms ) - ( 1 + 1 + 1 + ... n terms )

Since, ( 10 + 100 + 1000 + ... n terms ) forms an GP, we can write as,

⇒ 7/9 [( 10 ( 10ⁿ - 1  ) / 10 - 1 ] - ( 1 × n times )

⇒ 7/9 [ ( 10 ( 10ⁿ - 1 ) / 9 ] - n

⇒7/81 [ 10 ( 10ⁿ - 1 ) ] - n

This is the required answer !!

Thanks !!