Question

The sum of 5 numbers in an Arithmetic Sequence is 40 and the sum of their squares is 410. Find the numbers.​

1. The five numbers a-2d,a-d,a,a+d,a-2d is 2,5,8,11,14 and 14,11,8,5,2.
Given that the sum of 5 numbers in an Arithmetic Sequence is 40 and the sum of their squares is 410.
An arithmetic progression or arithmetic sequence (AP) is a sequence of numbers such that the difference between consecutive terms is constant.
Let the numbers be a-2d,a-d,a,a+d,a+2d.
It is given that their sum is 40.
So, we will add all the numbers and equate them with 40, we get
a-2d+a-d+a+a+d+a+2d=40
5a=40
a=8
It is also given that sum of the squares of the number is 410.
So, we will square and add all the numbers and equate them with 410, we get
(a-2d)²+(a-d)²+a²+(a+d)²+(a+2d)²=410
As we know that (a+b)²+(a-b)²=2(a²+b²).
By using this property, we get
2(a²+4d²)+2(a²+d²)+a²=410
As we find above that a=8.
Substitute this value, we get
2(64+4d²)+2(64+d²)+64=410
128+8d²+128+2d²+64=410
10d²+320=410
10d²=410-320
10d²=90
d²=9
d=±3
When a=8 and d=-3 the numbers are 14,11,8,5,2
When a=8 and d=3 the numbers are 2,5,8,11,14
Hence, the five numbers of arithmetic sequence when the sum of 5 numbers in an Arithmetic Sequence is 40 and the sum of their squares is 410 is 2,5,8,11,14 and 14,11,8,5,2.