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June 15th, 2018, 09:40 AM   #1
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Five numbers in arithmetic series

The sum of five numbers in arithmetic progression is 40 and the sum of their squares is 410. Find the five numbers.

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June 15th, 2018, 09:49 AM   #2
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$a_1 + (a_1+d) + (a_1+2d) + (a_1+3d) + (a_1+4d) = 40$

$a_1^2 + (a_1+d)^2 + (a_1+2d)^2 + (a_1+3d)^2 + (a_1+4d)^2 = 410$

solve the system for $a_1$ and $d$
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June 15th, 2018, 09:52 AM   #3
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Quote:
Originally Posted by Chanceux View Post
The sum of five numbers in arithmetic progression is 40
$k+(k+m)+(k+2m)+(k+3m)+(k+4m) = 40$


Quote:
and the sum of their squares is 410.
$k^2+(k+m)^2+(k+2m)^2+(k+3m)^2+(k+4m)^2 = 410$

Expand all this out into a linear equation and a quadratic equation which will have two solutions.
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Last edited by skipjack; June 15th, 2018 at 10:27 AM.
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June 15th, 2018, 09:59 AM   #4
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Math Focus: Calculus/ODEs
I would begin by writing:

$\displaystyle (a)+(a+d)+(a+2d)+(a+3d)+(a+4d)=40$

$\displaystyle 5a+10d=40$

$\displaystyle a+2d=8$

Then:

$\displaystyle (a)^2+(a+d)^2+(a+2d)^2+(a+3d)^2+(a+4d)^2=410$

This reduces to

$\displaystyle a^2+4ad+6d^2=82$

Now you have two equations and two unknowns...can you proceed?

edit: There were no replies when I started...sorry to be redundant.
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June 15th, 2018, 10:38 AM   #5
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$a^2 + 4ad + 6d^2 - (a + 2d)^2 = 82 - 8^2$, so $2d^2 = 18$.

It follows that the five numbers are 2, 5, 8, 11 and 14. The progression may consist of these numbers in ascending order or in descending order.
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June 15th, 2018, 11:59 AM   #6
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Great advice from all of you, guys! Big thanks to everyone.
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June 15th, 2018, 01:06 PM   #7
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Hey Chanceux, are you Lucky ?
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June 15th, 2018, 03:27 PM   #8
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$(a_3-2d) + (a_3-d) + a_3 + (a_3+d) + (a_3+2d) = 40$

$5a_3 = 40 \implies a_3 = 8$


$(a_3-2d)^2 + (a_3-d)^2 + a_3^2 + (a_3+d)^2 + (a_3+2d)^2 = 410$

$(8-2d)^2 + (8-d)^2 + 8^2 + (8+d)^2 + (8+2d)^2 = 410$

$(64 - 32d + 4d^2) + (64 - 16d + d^2) + 64 + (64+16d + d^2) +(64+32d+4d^2) = 410$

$320 + 10d^2 = 410$

$d^2 = 9 \implies d = \pm 3$

for $d = 3$ ... $\{ 2,5,8,11,14 \}$

for $d = -3$ ... $\{14,11,8,5,2 \}$
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