# Summation of 'n' terms of the given expression

• smart_worker
In summary, the conversation discusses finding the general formula to calculate the sum of a series, specifically the series of 1+11+111+1111+11111+... up to n terms. It is noted that the kth term of the series is ##\frac{10^k-1}9## which can be written as a summation ## \sum_{k = 1}^{n} \frac{10^k-1}9##. The conversation then moves on to discussing another series, 112+122+132+... and it is determined that the general term for this series is ##n^2## where n > 10 and n is a natural number. It is also noted
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) + ..........+ (10n-2 + 10n-2) + (10n-1)

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

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

smart_worker said:

## 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?

LCKurtz said:
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?

GENERAL TERM:n2 ,where n>10;nεN.

am i right?

smart_worker said:
in general how do you find the general term?

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:
$$\sum_{k = 11}^{\infty}k^2$$

Mark44 said:
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##.

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

## What is the formula for summation of 'n' terms of the given expression?

The formula for summation of 'n' terms of the given expression is: S = n/2 * (a + an), where 'n' represents the number of terms, 'a' represents the first term, and 'an' represents the last term.

## How do I find the value of 'n' in the given expression?

To find the value of 'n' in the given expression, you need to count the number of terms in the given expression. If the expression is in the form of a series or sequence, the value of 'n' will be the number of terms in that series or sequence.

## Can I use the formula for summation of 'n' terms for any expression?

Yes, the formula for summation of 'n' terms can be used for any expression as long as it follows a pattern and can be written in terms of 'n'.

## What does the value of 'S' represent in the formula for summation of 'n' terms?

The value of 'S' in the formula for summation of 'n' terms represents the sum of all the terms in the given expression. It is the final result of the summation.

## Are there any other methods for finding the summation of 'n' terms besides using the formula?

Yes, there are other methods for finding the summation of 'n' terms such as using mathematical properties like the sum of arithmetic or geometric progressions, or using computer software or programs designed for summation calculations.

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