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Question
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`
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