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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.

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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.

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$rightwards double arrow$ (a + bx) (b - cx) = (a - bx) (b + cx)

or ab - acx + b²x - bcx² = ab + cax - b²x - bcx²

or b²x + b²x = cax + acx

$rightwards double arrow$ 2b²x = 2acx or b² = ac or $straight b over straight a space equals space straight c over straight b$ ...(i)

Also, $fraction numerator straight b plus cx over denominator straight b minus cx end fraction space equals space fraction numerator straight c plus dx over denominator straight c minus dx end fraction space rightwards double arrow space space space left parenthesis straight b plus cx right parenthesis space left parenthesis straight c minus dx right parenthesis space equals space left parenthesis straight c plus dx right parenthesis left parenthesis straight b minus cx right parenthesis$

$rightwards double arrow$ $bc minus bdx plus straight c squared straight x minus cdx squared space equals space bc space minus space straight c squared straight x space plus space bdx space minus cdx squared$

$rightwards double arrow$ $space space straight c squared straight x plus straight c squared straight x space equals space bdx space plus space bdx space or space 2 straight c squared straight x space equals space 2 bdx$

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From (i) and (ii), we have

 $straight b over straight a space equals space straight c over straight d space equals space straight d over straight c$

Hence, a, b, c are in G.P.