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The sum of first three terms of a G.P. is 13/12 and their product is − 1. Find the G.P.
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Prove that:
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If sin α sin β − cos α cos β + 1 = 0, prove that 1 + cot α tan β = 0.
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If tan α = x +1, tan β = x − 1, show that 2 cot (α − β) = x2.
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The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is \[87\frac{1}{2}\] . Find them.
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If angle \[\theta\] is divided into two parts such that the tangents of one part is \[\lambda\] times the tangent of other, and \[\phi\] is their difference, then show that\[\sin\theta = \frac{\lambda + 1}{\lambda - 1}\sin\phi\]
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The sum of first three terms of a G.P. is \[\frac{39}{10}\] and their product is 1. Find the common ratio and the terms.
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If \[\tan\theta = \frac{\sin\alpha - \cos\alpha}{\sin\alpha + \cos\alpha}\] , then show that \[\sin\alpha + \cos\alpha = \sqrt{2}\cos\theta\].
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If α and β are two solutions of the equation a tan x + b sec x = c, then find the values of sin (α + β) and cos (α + β).
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The sum of three numbers in G.P. is 14. If the first two terms are each increased by 1 and the third term decreased by 1, the resulting numbers are in A.P. Find the numbers.
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The product of three numbers in G.P. is 216. If 2, 8, 6 be added to them, the results are in A.P. Find the numbers.
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Find three numbers in G.P. whose product is 729 and the sum of their products in pairs is 819.
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The sum of three numbers in G.P. is 21 and the sum of their squares is 189. Find the numbers.
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Find the sum of the following geometric progression:
2, 6, 18, ... to 7 terms;
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Find the sum of the following geometric progression:
1, 3, 9, 27, ... to 8 terms;
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Find the sum of the following geometric progression:
1, −1/2, 1/4, −1/8, ... to 9 terms;
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Find the sum of the following geometric progression:
(a2 − b2), (a − b), \[\left( \frac{a - b}{a + b} \right)\] to n terms;
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Find the sum of the following geometric progression:
4, 2, 1, 1/2 ... to 10 terms.
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Find the sum of the following geometric series:
0.15 + 0.015 + 0.0015 + ... to 8 terms;
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Find the sum of the following geometric series:
\[\sqrt{2} + \frac{1}{\sqrt{2}} + \frac{1}{2\sqrt{2}} + . . .\text { to 8 terms };\]
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