Question

Three numbers are in A.P have their sum 24. If first term is decreased by 1 and second term decreased by 2, then the new numbers are in G.P.Find the numbers.

Answer 1

Let the three terms be a−d,a,a+da−d,a,a+d where d is the common difference of the A.P.

Now the sum becomes

a−d+a+a+d=24a−d+a+a+d=24

3a=243a=24

a=8a=8

Also we’re given that,

a−d−1,a−2,a+da−d−1,a−2,a+d form a G.P

Putting the value of a,

7−d,6,8+d7−d,6,8+d form a G.P

So we’ve,

6=((7−d)(8+d))126=((7−d)(8+d))12

36=56−d−d236=56−d−d2

d2+d−20=0d2+d−20=0

On solving the given quadratic equation we get the values of d as

d=−5,4d=−5,4

So the possible terms are,

13,813,8 and 33 and 4,84,8 and 12.