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The sum or difference of two G.P.s, is again a G.P. - Mathematics

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MCQ
True or False

The sum or difference of two G.P.s, is again a G.P.

Options

  • True

  • False

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Solution

This statement is False.

Explanation:

Let us consider two G.P.’s

a1, a1r1, a1r12, a1r13 ... a1r1n-1

And a2, a2r2, a2r22, a2r23, ... a2r2n-1

Now Sum of two G.Ps

`(a_1 + a_2) + (a_1r_1 + a_2r_2) + (a_1r_1^2 + a_2r_2^2)  ...`

Now `T_2/T_1 = (a_1r_1 + a_2r_2)/(a_1 + a_2)`

And `T_3/T_2 = (a_1r_1^2 + a_2r_2^2)/(a_1r_1 + a_2r_2)`

But `(a_1r_1 + a_2r_2)/(a_1 + a_2) ≠ (a_1r_1^2 + a_2r_2^2)/(a_1r_1 + a_2r_2)`

Now let us consider the difference G.P’s

`(a_1 - a_2) + (a_1r_1 - a_2r_2) + (a_1r_1^2 - a_2r_2^2)`

∴ `T_2/T_1 = (a_1r_1 - a_2r_2)/(a_1 - a_2)`

And `T_3/T_2 = (a_1r_1^2 - a_2r_2^2)/(a_1r_1 - a_2r_2)`

But `T_2/T_1 ≠ T_3/T_2`

Concept: Geometric Progression (G. P.)
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APPEARS IN

NCERT Mathematics Exemplar Class 11
Chapter 9 Sequences and Series
Exercise | Q 33 | Page 164

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