Large Arithmetic Progression

Algebra Level 2

An arithmetic progression has one million terms. The last term is 2020 2020 and the common difference is 0.2 0.2 . What is the sum of the last 101 101 terms?


The answer is 203010.

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3 solutions

Richard Desper
Nov 12, 2020

The sum of an arithmetic progression can be viewed as the product of the number of terms with the median term. (Indeed, this framing is what gives us the general formula for such a sum.)

In the current problem, the number of terms is given to us: 101 101 . The median term can be calculated as 2020 50 ( 0.2 ) = 2010 2020 - 50*(0.2) = 2010 .

Thus the sum of the progression is 2010 101 = 203010 2010*101=203010 .

Chew-Seong Cheong
Nov 11, 2020

The sum of the last 101 101 terms of the arithmetic progression is given by:

S = 2020 + ( 2020 0.2 ) + ( 2020 2 ( 0.2 ) ) + + ( 2020 100 ( 0.2 ) ) = 2020 × 101 0.2 ( 1 + 2 + 3 + + 100 ) = 2020 × 101 1 5 ( 100 × 101 2 ) = ( 2020 10 ) 101 = 2010 × 101 = 203010 \begin{aligned} S & = 2020 + (2020 - 0.2) + (2020 - 2(0.2)) + \cdots + (2020 - 100(0.2)) \\ & = 2020 \times 101 - 0.2 (1+2 + 3 + \cdots + 100) \\ & = 2020 \times 101 - \frac 15 \left(\frac {100 \times 101}2\right) \\ & = (2020 - 10)101 = 2010 \times 101 = \boxed{203010} \end{aligned}

Hypergeo H.
Nov 10, 2020

Count backwards!
An AP in reverse is also an AP, with a common difference which is the negative of the common difference of the original AP.
Using ' to indicate the reverse AP, we have: a = 2020 , d = 0.2 S 101 = 101 2 ( 2 a + 100 d ) = 101 2 ( 2 ( 2020 ) + 100 ( 0.2 ) ) = 101 ( 2020 10 ) = 203010 \begin{aligned} a'&=2020, d'=-0.2\\ S'_{101}&=\frac {101}2(2a'+100d)\\ &=\frac {101}2 (2(2020)+100(-0.2))\\ &=101(2020-10)\\ &=203010 \end{aligned}

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