Q. 115.0( 1 Vote )

# The sum of the first three terms of a G.P. is 13 and sum of their squares is 91. Determine the G.P.

Answer :

Let the first term of G.P be a

⇒ second term = ar and third term = ar^{2}.

(where, r is the common ratio)

∵ sum of three terms is 13

⇒ a(1 + r + r^{2}) = 13r……..(1)

Also, sum of their squares is 91.

⇒ a^{2} (1 + r^{2} + r^{4}) = 91r^{2} ………(2)

Now, Squaring (1) dividing by (2)

⇒ 7( 1 + r^{2} + r) = 13( 1 + r^{2} - r)

⇒ (7 + 7r^{2} + 7r) = ( 13 + 13r^{2} - 13 r)

⇒ 6r^{2} - 20r + 6 = 0

⇒ 2(3r^{2} - 10r + 3) = 0

⇒ 3r^{2} - 10r + 3 = 0

⇒ 3r^{2} - 9r-r + 3 = 0

⇒ (3r – 1)(r – 3) = 0

Substituting r in equation (1), we get-

⇒a(1 + 3 + 9) = 13 × 3

And,

⇒13a = 13

And

⇒ a = 1 and a = 9.

Now, G.P is-

a, ar, ar^{2},….

⇒ If r = 3 and a = 1 then,

⇒1, 1×3, 1×3^{2}, ……

= 1, 3 , 9 are the first three terms.

And, If and a = 9 then,

= 9, 3, 1 are the first three terms.

⇒ 1,3,9,… … or 9,3,1,… … is the G.P.

Rate this question :

The fifth term of a G.P. is 1875. If the first term is 3, find the common ratio.

Tamil Nadu Board - MathFind out which of the following sequences are geometric sequences. For those geometric sequences, find the common ratio.

0.12, 0.24, 0.48,….

Tamil Nadu Board - MathWhich term of the geometric sequence,

(i) 5, 2, … , is (ii) 1, 2, 4, 8,…, is 1024 ?

Tamil Nadu Board - MathIf a, b, c, d are in a geometric sequence, then show that (a - b + c) (b + c + d) = ab + bc + cd.

Tamil Nadu Board - MathA company purchases an office copier machine for ₹50,000. It is estimated that the copier depreciates in its value at a rate of 15% per year. What will be the value of the copier after 15 years?

Tamil Nadu Board - MathIf ₹1000 is deposited in a bank which pays annual interest at the rate of 5% compounded annually, find the maturity amount at the end of 12 years.

Tamil Nadu Board - MathIf the third term of a G.P is 2, then the product of first 5 terms is

Tamil Nadu Board - MathThe common ratio of the G.P. a^{m-n}, a^{m}, a^{m + n}is

If x 0, then 1 + sec x + sec^{2}x + sec^{3}x + sec^{4}x + sec^{5}x is equal to