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Solved Examples based on GP

Solved examples :

Illustration:

The 7^th term of a G.P. is 8 times the 4^th term. Find the G.P. when its 5^th term is 48.

Solution:

Given that t₇ = 8t₄ ⇒ ar⁶ = 8ar³

⇒ r³ = 8 = 23 ⇒ r = 2.

Also t₅ = 48 Þ ar⁴ = 48 or 16a = 48 ⇒ a = 3.

Hence the required G.P. is 3, 6, 12, 24 ……

Illustration:

Does there exists a G.P. containing 27, 8 and 12 as three of its terms? If it exists, how many such progressions are possible?

Solution:

Let 8 be the m^th, 12 the n^th and 27 be the t^th terms of a G.P. whose first term is A and common ratio is R.

Then 8 = AR^m–1, 12 = AR^n–1, 27 = AR^t–1

⇒ 8/12 = R^m–n = 2/3, 12/27 = R^n–t = (2/3)², 8/27 = R^m–t =(2/3)³

⇒ 2m – 2n = n – t and 3m – 3n = m – t

⇒ 2m + t = 3n and 2m + t = 3n

⇒ 2m+t/3 = n.

There are infinite sets of values of m, n, t which satisfy this relation. For example, take m = 1, then 2+t/3 = n = k ⇒ n = k, t = 3k – 2. By giving different values to k we get integral values of n and t. Hence there are infinite numbers of G.P.’s whose terms may be 27, 8, 12 (not consecutive).

Illustration:

In a four term series if first three are in G.P. and last three are in A.P. with common different 6 and last terms is equal to the first term then find all four terms in series.

Solution:

This is very tricky question. If you read question carefully then it is clear that we have to start with A.P. because common difference is given.

Let the numbers be a + 6, a–6, a, a+6 now first three are in G.P. is (a–6)² = a(a+6) or, a² – 12a + 36 = a² therefore numbers are 8, –4, 2, 8.

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