Q. 33.9( 64 Votes )

# The 5^{th}, 8^{th} and 11^{th} terms of a G.P. are p, q and s, respectively. Show that q^{2} = ps.

Answer :

Given: 5^{th}, 8^{th} and 11^{th} terms of a G.P. are p, q and s, respectively

We know that in G.P a_{n} = ar^{n-1}

Here, n: number of terms

a: First term

r: common ratio

Here,

a_{5} = ar^{5-1} = ar^{4}

⇒ p = ar^{4} (∵ 5^{th} term of G.P. is given p) –1

Similarly,

a_{8} = ar^{8-1} = ar^{7}

⇒ q = ar^{7} (∵ 7^{th} term of G.P. is given q) –2

a_{11} = ar^{11-1} = ar^{10}

⇒ s = ar^{10} (∵ 11^{th} term of G.P. is given s) –3

We can observe that:

q × q = p × s (that is, ar^{7} × ar^{7} = ar^{4} × ar^{10})

∴ q^{2} = ps

Hence proved

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