Calvin learns summation!

Algebra Level 2

1 − 3 + 5 − 7 + 9 − 11 + 13 − 15 + … 1-3+5-7+9-11+13-15 + \ldots

Once Calvin learnt how to find sum of numbers from 1 to 100. So the curious calvin thought of alternating the signs in the summation as shown below. Find the value of first 100 terms in the summation above.

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The answer is -100.

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Nitesh Chaudhary by Nitesh Chaudhary , 27, India

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2 solutions

Feathery Studio
Jun 18, 2015

1 − 3 + 5 − 7 + 9 − 11 + 13 − 15 … … 1-3+5-7+9-11+13-15\dots \dots = ( 1 − 3 ) + ( 5 − 7 ) + ( 9 − 11 ) + ( 13 − 15 ) … … = (1-3)+(5-7)+(9-11)+(13-15)\dots \dots = ( − 2 ) + ( − 2 ) + ( − 2 ) + ( − 2 ) … … = (-2)+(-2)+(-2)+(-2)\dots \dots = ( − 2 ) × 100 2 = − 100 = (-2)\times\frac{100}{2} = \boxed{-100}

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Arunima Mitra
Sep 22, 2015

1-3+5-7+9-11+13... can be written as { (1+5+9+13+17...) - (3+7+11+15+19...) }

These are two Arithmetic Progression Series..

Since there are 100 terms in total, there are 50 terms in each series..

Sum1= (50/2) {2+ (49 x 4)} = 4950 Sum2= (50/2) {6+(49 x 4)} = 5050

Therefore Answer : 4950-5050 = -100

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Amazing Solution

Ajay Aggarwal - 5 years, 7 months ago

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