Find the sum to the n terms:
5+55+555+.......
1
1+11 = 12
1+11+111 = 123
So, for n terms, that ends up with
n*10^0 + (n-1)*10 + (n-2)*n^2
see whether you can arrive at the solution here:
http://www.wolframalpha.com/input/?i=sum(k%3D0..n-1)+(n-k)*10%5Ek
To find the sum to the nth terms of the given series, we can observe a pattern:
1st term = 5
2nd term = 5 * 10 + 5 = 55
3rd term = 5 * 100 + 55 = 555
From this pattern, we can deduce that the nth term is obtained by multiplying the previous term by 10 and adding 5.
Now, let's find a general formula for the nth term of the series. We can express it as:
nth term = 5 * (10^(n-1)) + previous term
Using this formula, we can calculate the sum of the series up to the nth term using the formula for the sum of a geometric series:
Sum = (first term * (1 - common ratio^n)) / (1 - common ratio)
In our case, the first term is 5, and the common ratio is 10. So, substituting these values into the formula, we have:
Sum = (5 * (1 - 10^n)) / (1 - 10)
Simplifying further:
Sum = (5 * (1 - 10^n)) / (-9)
Therefore, the sum to the nth terms of the series 5+55+555+... is (5 * (1 - 10^n)) / (-9).