Question

5. Find five numbers in A.P. whose sum is 25/
2
and the ratio of the first to the last terms is
2:3 and also find its first term(a) and Common difference

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Answer 1

Answer:-

Let the numbers be a - 2d , a - d , a , a + d , a + 2d .

Given:

Sum of the numbers = 25/2

→ a - 2d + a - d + a + a + d + a + 2d = 25/2

→ 5a = 25/2

→ 5 * a = 25/2

→ a = (25/2) * (1/5)

→ a = 5/2

Also,

Ratio of first term to the last term = 2 : 3

→ a - 2d : a + 2d = 2 : 3

Putting the value of a we get,

→ 5/2 - 2d : 5/2 + 2d = 2 : 3

→ (5 - 4d)/2 : (5 + 4d) / 2 = 2 : 3

→ [ (5 - 4d)/2 ] * [ 2/(5 + 4d) ] = 2/3

→ (5 - 4d) / (5 + 4d) = 2/3

On cross multiplication we get,

→ 3 * (5 - 4d) = 2 * (5 + 4d)

→ 15 - 12d = 10 + 8d

→ 15 - 10 = 8d + 12d

→ 5 = 20d

→ 5/20 = d

→ 1/4 = d

Hence,

• a - 2d = 5/2 - 2(1/4) = (5 - 1)/2 = 4/2 = 2

• a - d = 5/2 - 1/4 = (10 - 1)/4 = 9/4.

• a = 5/2

• a + d = 5/2 + 1/4 = (10 + 1)/4 = 11/4

• a + 2d = 5/2 + 2(1/4) = (5 + 1)/2 = 6/2 = 3.

Therefore,

• First term (a) = 5/2

• Common difference (d) = 1/4

• Five numbers are 2 , 9/4 , 5/2 , 11/4 , 3.