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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.

   Learn more about the arithmetic sequence from here brainly.com/question/503167

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