Question

Find the value of :
100² - 99² +98² - 97² +....+ 2²-1²
(1) 5010
(2) 5040
(3) 5050
(4) 4050​

Answer 1

Answer:

Given series is 100^2 - 99^2 + 98^2 -97^2 + 96^2 - 95^2+ - - - - -+ 2^2 - 1^2

Now we all know the formaula - (a^2 - b^2) = (a+b)(a-b)

You need to us this formula to find the sum of the series and we will do this by finding all pairs of (a^2 - b^2) in the series.

And they are (100^2 - 99^2), (98^2 - 97^2), (97^2 - 96^2), … , (2^2 - 1^2)

Apply the formula on each pair and the series becomes

(100 + 99)(100 - 99) + (98 + 97)(98 - 97) + (97 + 96)(97 - 96) + - - - - + (2+1)(2 - 1)

Now all the (a-b) terms here becomes 1 ; example - (100 -99) = 1, (98 - 97) = 1, (97 - 96) = 1 and so on.

So only (a + b) terms are left in the series and the series becomes -

(100 + 99) + (98 + 97) + (96 + 95) + - - - - +(2 + 1)

This becomes an AP where first term is 100 and common difference is -1.

Apply the formula to find sum of terms of an AP i.e.

Sum = (n/2)[2a + (n - 1)*d]

Sum = (100/2)[2*100 + (100 - 1) *(-1)]

Solve it and you will get the sum as 5050.

Hope it helps!!

Answer 2

Answer:

answer is 5050

Step-by-step explanation:

Given series is 100^2 - 99^2 + 98^2 -97^2 + 96^2 - 95^2+ - - - - -+ 2^2 - 1^2

Now we all know the formaula - (a^2 - b^2) = (a+b)(a-b)

You need to us this formula to find the sum of the series and we will do this by finding all pairs of (a^2 - b^2) in the series.

And they are (100^2 - 99^2), (98^2 - 97^2), (97^2 - 96^2), … , (2^2 - 1^2)

Apply the formula on each pair and the series becomes

(100 + 99)(100 - 99) + (98 + 97)(98 - 97) + (97 + 96)(97 - 96) + - - - - + (2+1)(2 - 1)

Now all the (a-b) terms here becomes 1 ; example - (100 -99) = 1, (98 - 97) = 1, (97 - 96) = 1 and so on.

So only (a + b) terms are left in the series and the series becomes -

(100 + 99) + (98 + 97) + (96 + 95) + - - - - +(2 + 1)

This becomes an AP where first term is 100 and common difference is -1.

Apply the formula to find sum of terms of an AP i.e.

Sum = (n/2)[2a + (n - 1)*d]

Sum = (100/2)[2*100 + (100 - 1) *(-1)]

Solve it and you will get the sum as 5050.

Hope it helps!!

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