Summation of 'n' terms of the given expression

[smart_worker]

Homework Statement

find the general formula to calculate the sum

Homework Equations

1+11+111+1111+11111+....upto n terms

The Attempt at a Solution

10⁰ + (10¹+10⁰) + (10²+10¹ + 10⁰) + (10³ + 10²+10¹ + 10⁰) + ...

==> (10⁰+10⁰+10⁰+...upto n terms) + (10¹+10¹+10¹+...upto n-1 terms) + (10²+10²+10²+...upto n-2 terms) + ..........+ (10^n-2 + 10^n-2) + (10^n-1)


==>n + (10¹+10¹+10¹+...upto n-1 terms) + (10²+10²+10²+...upto n-2 terms) + ......+ (10^n-2 + 10^n-2) + (10^n-1)


after this i don't know how.They seem to resemble geometric series.

 

[LCKurtz]

Homework Statement

find the general formula to calculate the sum

Homework Equations

1+11+111+1111+11111+....upto n terms

Notice that the kth term of that is $\frac{10^k-1}9$. Does that help?

 

[smart_worker]

Notice that the kth term of that is $\frac{10^k-1}9$. Does that help?

in general how do you find the general term?

how about this series:11²+12²+13²+...
GENERAL TERM:n² ,where n>10;nεN.

am i right?

 

[Mark44]

in general how do you find the general term?

how about this series:11²+12²+13²+...
GENERAL TERM:n² ,where n>10;nεN.

am i right?

Yes, that's the general term. The series can be written as a summation like so:
$$ \sum_{k = 11}^{\infty}k^2$$

 

[Ray Vickson]

Yes, that's the general term. The series can be written as a summation like so:
$$ \sum_{k = 11}^{\infty}k^2$$

This is horribly divergent. However, $\sum_{k=11}^N k^2$ is meaningful for any finite $N$.

 

[Chestermiller]

In your first post, you noted that each term seems to resemble a geometric series. Not only does it seem to resemble a gerometric series. That's exactly what each of the terms is. The sum of a geometric series is (arⁿ-a)/(r-1). In your case, a = 1 and r = 10. That's how to get the general term that LCKurtz presented.

Chet