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Question

Find the sum of 12-22+32-42+52-62.......?


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Solution

Step 1: Calculate the sum of the series, if the number of terms is even

Given series: 12-22+32-42+52-62+.......

Let the number of terms of the series be n, which is an even number.

Thus, 12-22+32-42+52-62.......+n-12-n2

=1-21+2+3-43+4+5-65+6+.......+n-1-nn-1+n=-11+2+-13+4+-15+6+......+-1n-1+n=-11+2+3+4+5+6+.....+n

Use the formula of the sum of the first n natural numbers.

=-1nn+12=-n2+n2

Hence, the sum of the series, when the number of terms is even, is -(n2+n2).

Step 2: Calculate the sum of the series, if the number of terms is odd:

Given series: 12-22+32-42+52-62.......

Let the number of terms of the series be n, which is an odd number.

Thus, 12-22+32-42+52-62.......+n-22-n-12+n2

=1-21+2+3-43+4+5-65+6+.......+n-2-n-1n-2+n-1+n2=-11+2+-13+4+-15+6+......+-1n-2+n-1+n2=-11+2+3+4+5+6+.....+n-1+n2

Step 3: Calculate the sum of the first n natural number

Use the formula of the sum of the first n natural numbers.

=-1n-1n-1+12+n2=-n-1n2+n2=-n2-n2+n2=-n2+n+2n22=n2+n2

Hence, the sum of the series, when the number of terms is odd, is n2+n2.

Thus, in general, the sum of the series can be calculated by -1n+1n2+n2

Therefore, the sum of the given series is -1n+1n2+n2, where n is an integer.


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