Verified answer

i) S = 4 + 44 + 444 + 4444 + ------- + 444444444

ii) Taking out 4, S = 4(1 + 11 + 111 + 1111 + ------- + 111111111)

==> S = (4/9)(9 + 99 + 999 + 9999 + ------- + 999999999) [Multiplying & dividing by 9]

==> S = (4/9)[(10 - 1) + (100 - 1) + (1000 - 1) + (10000 - 1) + --------- +(1000000000 - 1)]

==> S = (4/9)[(10 + 100 + 1000 + 10000 + ----- + 1000000000) - (1 + 1 + 1 + --)]

In the above,

10 + 100 + 1000 + ---- is in GP, with a = 10; Common ratio r = 10 and no. of terms = 9

So its sum = 10(10^9 - 1)/9 = (10/9)(999999999) = 1111111110

1 + 1 + 1 + ---- = 9

Thus S = (4/9)[1111111110 - 9] = (4/9)(1111111101) 4*123456789 = 493827156

Note:

If in case you require S = 4 + 44 + 444 + ---- to n terms, then

S = (4/9)[(10/9){10^n - 1} - n]

4+

44+

444+

4444+

44444+

444444+

4444444+

44444444+

444444444 =

4, 44, 444, 4444, ....

an = 4/9(10^n - 1)

4 + 44 + 444 + 4444 + ... + 444444444

Sum of first 3 terms: 492

Sum of next 3 terms: 493,332

Sum of next 3 terms: 493,333,332

Total sum: 493,827,156

7

what means dots

i dont understand the dots?

493827156