Question

Find 3 nos in gp whose sum is 13 and sum of squares is 91

Answer 1

Let 3 numbers say a, b, c  be in GP

then we know b² = a.c

also given  a+b+c = 13

⇒ (a+b+c)² = 13²

⇒ a² +  b² + c² + 2a.b + 2b.c +2a.c = 169

also given that  a² + b² + c² = 91 subst that in abv eqn

91 + 2ab + 2bc + 2ca = 169

2 (ab + bc + ac) = 78

ab + bc + ca = 39

ab + bc + b² = 39

b( a + b + c) = 39

b (13) = 39

b = 3

b² = a.c

a.c = 9

also a + c = 10 ⇒ c = 10 - a

a(10 - a) = 9

10.a - a² = 9

a² - 10.a + 9 = 0

so a₁ = 9 and a₂ = 1

if a₁ = 9 then c₁ = 1

else if a₂ = 1 then c₂ = 9

so the GP₁ = 1, 3, 9

or GP₂ = 9, 3, 1

can be either of them

Hope that helped

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