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QUESTION: 1

In a G.P. if the (p+q)^{th} term is m and the (p-q)^{th} term is n then the pth term is _________.

Solution:

QUESTION: 2

The sum of n terms of the series 0.5+0.05+0.555+………… is

Solution:

QUESTION: 3

If a, b-a, c-a are in G.P. and a=b/3=c/5 then a, b, c are in

Solution:

QUESTION: 4

The sum of term of the series 5+55+555+…..is

Solution:

QUESTION: 5

If a, b, (c+1) are in G.P. and a=(b-c)^{2} then a, b, c are in

Solution:

QUESTION: 6

If S_{1}, S_{2}, S_{3}, ………S_{n} are the sums of infinite G.P.s whose first terms are 1, 2, 3……n and whose common ratios are 1/2, 1/3, ……1/(n+1) then the value of S_{1}+S_{2}+S_{3}+ ……S_{n} is

Solution:

QUESTION: 7

If a, b, c are in G.P. then the value of (a-b+c)(a+b+c)^{2}-(a+b+c)(a^{2}+b^{2}+c^{2}) is given by

Solution:

QUESTION: 8

If a^{1/x}=b^{1/y}=c^{1/z} and a, b, c are in G.P. then x, y, z are in

Solution:

QUESTION: 9

If a, b, (c+1) are in G.P. and a=(b-c)^{2} then a, b, c are in

Solution:

QUESTION: 10

If a, b, c are in G.P. then a^{2}+b^{2}, ab+bc, b^{2}+c^{2} are in

Solution:

QUESTION: 11

If a, b, c are in G.P then the value of a (b^{2}+c^{2})-c(a^{2}+b^{2}) is given by

Solution:

QUESTION: 12

If a, b, c are in G.P. then value of a^{2}b^{2}c^{2}(a^{-3}+b^{-3}+c^{-3})-(a^{3}+b^{3}+c^{3}) is given by

Solution:

QUESTION: 13

If a, b, c, d are in G.P. then a+b, b+c, c+d are in

Solution:

QUESTION: 14

If a, b, c are in A.P. a, x, b are in G.P. and b, y, c are in G.P. then x^{2}, b^{2}, y^{2} are in

Solution:

QUESTION: 15

If a, b, c, d are in G.P. then the value of (b-c)^{2}+(c-a)^{2}+(d-b)^{2}-(a-d)^{2} is given by

Solution:

QUESTION: 16

If (a-b), (b-c), (c-a) are in G.P. then the value of (a+b+c)^{2}-3(ab+bc+ca) is given by

Solution:

QUESTION: 17

If a, b, c, d are in G.P. then (a-b)^{2}, (b-c)^{2},(c-a)^{2} are in

Solution:

QUESTION: 18

If a, b, x, y, z are positive numbers such that a, x, b are in A.P. and a, y, b are in G.P. and z=(2ab)/(a+b)then

Solution:

QUESTION: 19

If a, b-a, c-a are in G.P. and a=b/3=c/5 then a, b, c are in

Solution:

QUESTION: 20

If a, b, c, d are in G.P. then the value of b(ab-cd)-(c+a)(b^{2}-c^{2}) is ________

Solution:

QUESTION: 21

The least value of n for which the sum of n terms of the series 1+3+3^{2}+………..is greater than 7000 is _________.

Solution:

QUESTION: 22

If a, b, c are in A.P. and x, y, z in G.P. then the value of (x^{b}.y^{c}.z^{a}) ÷ (x^{c}.y^{a}.z^{b}) is _________.

Solution:

QUESTION: 23

If a, b, c are the p^{th}, q^{th} and r^{th} terms of a G.P. respectively the value of a^{q-r}. b^{r-p}. c^{p-q} is ___________.

Solution:

QUESTION: 24

If a, b, c are in G.P. then the value of a(b^{2}+c^{2})-c(a^{2}+b^{2}) is __________

Solution:

QUESTION: 25

If S be the sum, P the product and R the sum of the reciprocals of n terms in a G.P. then P is the ________ of S^{n} and R-^{n}.

Solution:

QUESTION: 26

If the sum of three numbers in G.P. is 21 and the sum of their squares is 189 the numbers are __________.

Solution:

QUESTION: 27

The sum of n terms of the series 7+77+777+……is

Solution:

QUESTION: 28

If a, b, c, d are in G.P. then the value of (ab+bc+cd)^{2}-(a^{2}+b^{2}+c^{2})(b^{2}+c^{2}+d^{2})is ______.

Solution:

QUESTION: 29

If 1+a+a^{2}+………∞=x and 1+b+b^{2}+……∞=y then 1+ab+a^{2}b^{2}+………∞ = x is given by ________.

Solution:

**ANSWER :- a**

**Solution :- Given, x=1+a+a^2+......∞**

**Since this is a infinite G.P. series, where, (first term)=1 and (common difference)=a,**

**So, x = 1/(1−a)**

**⇒ x−ax=1**

**⇒ ax=x−1**

**⇒ a=(x−1)/x**

**Similarly, y=1+b+b^2 +......∞ is a infinite G.P. series, where, (first term)=1 and**

** (common difference)=b,**

**So, y = 1/(1−b)**

**⇒ y−by=1**

**⇒ by=y−1**

**⇒ b=(y−1)/y**

**And now,**

**L.H.S.=1 + ab + a^2b^2 + ....∞**

**= 1/(1−ab) (infinte G.P. series where (first term)=1 and (common difference)=ab**

**= 1/{1−(x−1/x)(y−1/y)}**

** = xy/(xy−xy+x+y−1)**

**= (xy)/(x+y−1)**

QUESTION: 30

Sum upto ∞ of the series 1/2+1/3^{2}+1/2^{3}+1/3^{4}+1/2^{5}+1/3^{6}+……is

Solution:

QUESTION: 31

If a, b, c are in A.P. and x, y, z in G.P. then the value of x^{b-c}. y^{c-a}. z^{a-b} is ______.

Solution:

QUESTION: 32

If the sum of three numbers in G.P. is 35 and their product is 1000 the numbers are _________.

Solution:

QUESTION: 33

Three numbers whose sum is 15 are A.P. but if they are added by 1, 4, 19 respectively they are in G.P. The numbers are _________.

Solution:

Let the given numbers in A.P. be *a* – *d*, *a*, *a *+ *d*.

According to question,

Hence, the numbers are 5 – *d*, 5, 5 + *d*.

Adding 1, 4 and 19 in first, second and third number respectively, we get

Since these numbers are in G.P.

Hence the numbers are

26, 5, –16 or 2, 5, 8.

QUESTION: 34

n(n-1)(2n-1) is divisible by

Solution:

For n=1

n(n+1)(2n+1) = 6, divisible by 6.

Let the result be true for n=k

Then, k(k+1)(2k+1) is divisible by 6.

So k(k+1)(2k+1) =6m (1)

Now to prove that the result is true for n=k+1

That is to prove, (K+1)(k+2)(2k+3) is divisible by 6.

(K+1)(k+2)(2k+3)=(k+1)k(2k+3)+(k+1)2(2k+3)=(k+1)k(2k+1)+(k+1)k2+(k+1)2(2k+3)

=6m+2(k+1)(k+2k+3) using (1)

=6m+2(k+1)(3k+3)

=6m +6(k+1)(k+1)=6[m+(k+1)^2]

So divisible by6.

QUESTION: 35

The value of n^{2}++2n[1+2+3+…+(n-1)] is

Solution:

QUESTION: 36

The sum of n terms of the series 1^{3}/1+(1^{3}+2^{3})/2+(1^{2}+2^{2}+3^{3})/3+……is

Solution:

QUESTION: 37

3^{n}-2n-1 is divisible by

Solution:

QUESTION: 38

The sum of n terms of the series 3+6+11+20+37+……… is

Solution:

QUESTION: 39

The nth terms of the series is 1/(4.7)+1/(7.10)+1/(10.13)+………is

Solution:

QUESTION: 40

If the third term of a G.P. is the square of the first and the fifth term is 64 the series would be __________.

Solution:

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