three numbers are in A P and their sum is 15 , if 1, 4 and 19 are added - Maths - Sequences and Series

Question

three numbers are in A P and their sum is 15 , if 1, 4 and 19
are added to these numbers respectively the resulting numbers are
in G P find the numbers

Shridhar Kaushik · Accepted Answer

Let the given numbers in A P be a â€“ d, a, a + d According to question, $(a - d) + a + (a + d) = 15 \Rightarrow 3 a = 15 \Rightarrow a = 5$ Hence, the numbers are 5 â€“ d, 5, 5 + d Adding 1, 4 and 19 in first, second and third number respectively, we get $6 - d, 9, 24 + d$ Since these numbers are in G P $∴ (9)^{2} = (6 - d) (24 + d) \Rightarrow 81 = 144 - 18 d - d^{2} \Rightarrow d^{2} + 18 d - 144 + 81 = 0 \Rightarrow d^{2} + 18 d - 63 = 0 \Rightarrow d^{2} + 21 d - 3 d - 63 = 0 \Rightarrow d (d + 21) - 3 (d + 21) = 0 \Rightarrow (d + 21) (d - 3) = 0 \Rightarrow d = - 21 or d = 3$ Hence the numbers are 26, 5, â€“16 or 2, 5, 8

three numbers are in A.P. and their sum is 15 , if 1, 4 and 19 are added to these numbers respectively the resulting numbers are in G.P.. find the numbers.