Q. 14.9( 7 Votes )

Three numbers are in A.P. Their sum is 27 and the sum of their squares is 275. Find the numbers.

Answer :

Let the three numbers are in AP = a, a + d, a + 2d


According to the question,


The sum of three terms = 27


a + (a + d) + (a + 2d) = 27


3a + 3d = 27


a + d = 9


a = 9 – d …(i)


and the sum of their squares = 275


a2 + (a + d)2 + (a + 2d)2 = 275


(9 – d)2 + (9)2 + ( 9 – d + 2d)2 = 275 [from(i)]


81 + d2 – 18d + 81 + 81 + d2 + 18d = 275


243 + 2d2 = 275


2d2 = 275 – 243


2d2 = 32


d2 = 16


d = √16


d = ±4


Now, if d = 4, then a = 9 – 4 = 5


and if d = – 4, then a = 9 – ( – 4) = 9 + 4 = 13


So, the numbers are


if a = 5 and d = 4


5, 9, 13


and if a = 13 and d = – 4


13, 9, 5


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