Question

7+77+777+.....+ upto n terms

Answer 1

Answer:

Step-by-step explanation:

Hi, my friend

7 + 77 + 777 + ....n terms

We can take 7 in common

=> (7)(1 + 11 + 111 + ....n terms)

Multiplying and dividing my any number doesn't change the value

So we multiply and divide by 9. Because we are multiplying 9/9 which is multiplied by 1.

=> (9/9)(7)(1 + 11 + 111 + ....n terms))

Take the numerator 9 inside.

=>(1/9)(7)(9 + 99 + 999 +....n terms))

=> (7/9)(9 + 99 + 999 +....n terms)

9 can be written as (10 - 1)

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

Now we can separate 10s and 1s

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

Terms of 10s can be written in form of powers.

And also sum of 1s till n = Σ1 = n

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

Now 10s are in Geometric Progression.

r = 100/10 = 10

Hence a = 10, r = 10, n = n

Substitute in the equation (2)

Sum of a Geometric series formula:

$\frac{a(r^{n}-1)}{r - 1}$ ---------------(2)

The below equals the Geometric series sum.

$\frac{10(10^{n}-1)}{10-1} \\\\=> \frac{10(10^{n}-1)}{9}$

Hence our original equation i.e. (1) equation deduces to

=> (7/9)(($\frac{10(10^{n}-1)}{9}$) - n)

And hence us our answer.

Harith

Maths Aryabhatta

Answer 2

answer is in attachment

thank. you