Five numbers form an arithmetic sequence with a mean of 18. if the mean of the squares of the five numbers is 374, what is the greatest of the five original numbers?

Let the five numbers which form arithmetic sequence be a-2d,a-d,a,a+d,a+2d where a is the first term and d is common difference.

Mean of these 5 numbers= 18

→ a-2d+a-d+a+a+d+a+2d = 5×18

→ 5 a= 5×18

Dividing both side by 5, we get , →a = 18

It is also given Mean of the squares of the five numbers is 374.

(a- 2 d)² + (a- d)² + a² +(a +d )² +(a+ 2 d)² = 5 × 374

→ a² + 4 d²- 4 ad +a² +d² -2 a d +a² +a² +d² +2 a d+a² + 4 d²+ 4 ad =5 × 374

→ 5 a² + 10 d²=5 × 374

→ 5 × (a² + 2 d²)=5 × 374

Dividing both sides by 5,we get

a² + 2 d²= 374

As , a=18, Substituting the value of a in above equation

→18² + 2 d²=374

→324 + 2 d²=374

→ 2 d² = 374 -324

→ 2 d²=50

Dividing both side by 2, we get

→ d²= 25

→ d² =5²

→ d = 5

The five numbers are, 18-2×5, 18 -5,18,18+5, 18 +2×5 ,= 8, 13, 18, 23, 28.

So, greatest number among these 5 numbers are 28.

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