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Sequences and Series

Short Answer Type

61.

The fourth term of a GP. is 27 and the 7th term is 729, find the GP.

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62.
The seventh term of a GP. is 8 times the fourth term and 5th term is 48. Find the GP.

Let a be the first term and r be the common ratio.

According to the given condition,

                         $space space space space straight t subscript 7 space equals space 8 straight t subscript 4 space space space rightwards double arrow space space ar to the power of 7 minus 1 end exponent space equals space 8 ar to the power of 4 minus 1 end exponent$

$rightwards double arrow$                         $ar to the power of 6 space equals space 8 ar cubed space space space space space space rightwards double arrow space space straight r cubed space equals space 8 space space space space space space space space space space space space space space space space rightwards double arrow space space space space space space space space space straight r space equals space 2$

Also,                          $straight t subscript 5 space equals space 48 space space space space space space space rightwards double arrow space space ar to the power of 5 minus 1 end exponent space equals space 48 rightwards double arrow space ar to the power of 4 space equals space 48 space rightwards double arrow space straight a left parenthesis 2 right parenthesis to the power of 4 space equals space 48 space rightwards double arrow space space straight a space equals space 48 over 16 space equals space 3$

∴ G.P is 3, 3(2), 3(2)², ....... or 3, 6, 12,.......

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63.

If the GP.’s 5,10, 20, .......and 1280, 640, 320, ......have their nth terms equal, find the value of n.

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64.

The third term of a GP. is 4. Find the product of its first five terms.

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65.

The 5th, 8th and 11th terms of a GP. are p, q and s respectively. Show that q² = ps.

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66.

Find all the sequences which are simultaneously in A.P. and GP.

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67.

In a finite GP. the product of the terms equidistant from the beginning and the end is always same and equal to the product of first and last term.

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68.

If the first and the nth terms of a GP. are a and b respectively and if P is the product of first n terms, prove that P² = (ab)ⁿ.

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Long Answer Type

69.

If $fraction numerator straight a plus bx over denominator straight a minus bx end fraction space equals space fraction numerator straight b plus cx over denominator straight b minus cx end fraction space equals space fraction numerator straight c plus dx over denominator straight c minus dx end fraction$ $left parenthesis straight x not equal to 0 right parenthesis comma space$ show that a, b, c, d are in G.P.