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Syllabus

if a

^{2}, b^{2},c^{2 }are in a A.P .then prove that the following are also in A.P (i) 1/b+c ,1/c+a ,1/a+b (ii) a/b+c , b/a+c ,c/b+aIf a+b+c not equal to zero and b+c/a,c+a/b,a+b/c are in AP.Prove that 1/a,1/b,1/c are also in AP

Show that a^2,b^2,c^2 are in AP if and only iff 1/b+c,1/c+a,1/a+b are in AP

{Experts can you please clarify my doubts}

Find the sum to n terms of the series :- 5 + 11 + 19 + 29 + 41 +..........

Find the sum of integers from 1 to 100 that are divisible by 2 or 5.

$89.\mathrm{If}{\mathrm{a}}_{1},{\mathrm{a}}_{2},{\mathrm{a}}_{3},{\mathrm{a}}_{4}\mathrm{are}\mathrm{in}\mathrm{H}.\mathrm{P}.,\mathrm{then}\frac{1}{{\mathrm{a}}_{1}{\mathrm{a}}_{4}}\sum _{\mathrm{r}=1}^{3}{\mathrm{a}}_{\mathrm{r}}{\mathrm{a}}_{\mathrm{r}+1}\mathrm{is}\mathrm{a}\mathrm{root}\mathrm{of}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(\mathrm{A}\right){\mathrm{x}}^{2}+2\mathrm{x}+15=0\left(\mathrm{B}\right){\mathrm{x}}^{2}+2\mathrm{x}-15=0\phantom{\rule{0ex}{0ex}}\left(\mathrm{C}\right){\mathrm{x}}^{2}-6\mathrm{x}-8=0\left(\mathrm{D}\right){\mathrm{x}}^{2}-9\mathrm{x}+20=0\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

a thief runs away from a police station with a uniform speed of100 m per min. after a minute a policeman runs behind him to catch. he goes at a speed of 100 m in 1

^{st }min and increases his speed by 10 m each succeeding min. after how many min, the policeman catch the thief?If the pth, qth and rth terms of a GP be a, b, c respectively, prove that

a^(q-r).b^(r-p).c^(p-q)=1;

where ^=raised to the power

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

( answer = 1024 )

insert three numbers between 3 and 19 such that the resulting sequence is an ap

If a2, b2, c2, are in AP, then show that a/b+c, b/c+a, c/a+b are in AP.

In an A.P.,s

_{3}=6,s_{6}=3,prove that 2(2n+1)S_{n}_{+4}=(n+4)S_{2n+1}.if (m+1)th term of an AP is twice the n+1th term, prove that (3m+1)th term is twice the (m+n+1)th term.

please answer ASAP, exam tmrw.

the sum of first n terms of the series 1square + 2* 2square + 3square +2*4square + 5square + 2*6square +... is n*(n+1)square/2 when n is even. What is the sum when is odd?

If 7 times the 7th term of an A.P. is equal to 11 times the 11th term,show that 18th term of A.P. is 0.

The sum of 11 terms of an AP whose middle term s 30 is

(1) 320 (2)330 (3)340 (4)350

If the sum of n terms of an A.P. is 2mn + pn

^{2}, where m and p are constants, find the common differenceSum the series: 1+3+7+15+31+............to n terms.

If 1/x,1/y,1/z are AP, show that:

(i) yz,zx,xy are in AP

(ii) xy,zx,yz are in AP

(iii) y+z/x,z+x/y,x+y/z are in AP

I got the first one but how to do the other two?

If a, b , c are in G.P, and x, y be the arithmetic means of a, b and b, c respectively, prove that (i) a/x+c/y=2 (ii)1/x + 1/y= 2/b

If a,b,c are in A.P. and a,b,d are in G.P. ,prove that a,a-b,d-c are in G.P.

If first term ,second , and last term of an A.P be a ,b , c, respectively . Then show that the sum is

(b+c-2a) (a+c) / 2(b-a) .

the first term of the GP is 85, the sum of second term of AP and the second term of the GP is

76 and that of the 3rd term of AP and 3rd term of GP is 84. The sum of the AP is 126. Find each

term of AP and GP

how many terms of the sequence 18,16,14......should be taken so that their sum is zero.

^{3}+ 19/6^{3}+ 37/12^{3}+61/20^{3 }+_{............ }infinity .if p

^{th}, q^{th}, r^{th}and s^{th}terms of an A.P. are in G.P.,show that (p-q),(q-r) and (r-s) are also in G.P.If

a,b,c,dare in G.P, prove that are in G.P.If a

^{2}+2bc,b^{2}+2ac,c^{2}+2ab are in A.P. ,show that 1/(b-c),1/(c-a),1/(a-b) are also in A.P.find the sum of integers from 1 to 100 that are divisible by 2 or 5?

calculate the least no of terms of gp 5+10+20+...whose sum would exceed1000000

prove that:2^1/4 . 4^1/8 . 8^1/16............upto infinity=2

(where summation is from n=1 to n=20)

The sum of three numbers in G.P. is 21 and the sum of their squares is 189. Find the numbers.

let x=1+a+a2+.............. y=1+b+b2+.......... thn prove that

1+ab+a2b2+.......=xy/x+y-1

if p arithmetic means are inserted between a and b, prove that d=b-a/p+1

If the p

^{th},q^{th}terms of a G.P. are q,p respectively ,show that (p+q)^{th}term is (q^{p}/p^{q})^{1/p-q}^{b-c})x (Y^{c-a}) x(Z^{a-b})=1_{p+q}=m and t_{p-q}=n. Then find t_{p}.Find the sum of all natural numbers lying between 100 and 1000, which are multiples of 5.

In the arithmetic progression 2,5,8....upto 50 terms,and 3,5,7,9....upto 60 terms ,find how many pairs are identical and find the identical pairs.

Find the sum to

nterms of the sequence, 6, 66, 666, 6666…S1,S2,S3,...Sn are sums of n infinite geometric progressions. The first terms of these progressions are 1,2

^{2}-1,2^{3}-1,...2^{n}-1 and the common ratios are 1/2,1/2^{2},1/2^{3}...1/2^{n}. Calculate the value of S1+S2+...+SnIf 1/a, 1/b, 1/c are in AP, prove that (i) b + c / a, c +a / b, a + b / c are in AP (ii) a(b +c), b (c +a), c ( a + b) are in AP

find the sum of the first 25 trrms of the series 5.6+6.7+7.8+.....

If in an A.P., S

_{n}= n^{2}p and S_{m}=m^{2}p, then S_{p}=?1+(1+2)+(1+2+3)+(1+2+3+4)... find the sum of the series

if AM between two numbers exceeds their GM by 2 and the ratio of two numbers is 4, find the numbersa) 1/2 (root 5) b) root 5 c) 1/2 (root 5 - 1) d) 1/2 (1- root 5 )

_{k}a_{k-1 }+ a_{k-1}a_{k-2} = 2a_{k}a_{k-2 }, k>=3 and a_{1}=1 , here S_{p}= sigma (k = 1 to k = p) 1/a_{k }and given that S_{2p}/S_{p }does not depend on p then 1/a_{2016 }is = ?FIND FOUR NUMBERS IN AP WHOSE SUM IS 20 &SUM OF WHOSE SQUARES IS 120

Find two numbers whose difference is 2 and whose AM exceeds the GM by 1/2.The difference between any two consecutive interior angles of a polygon is 5°.If the smallest angle is 120°, find the number of the sides of the polygon.

sum to n terms

three numbers are in A.P. and their sum is 15 , if 1, 4 and 19 are added to these numbers respectively the resulting numbers are in G.P.. find the numbers.

The sum of an infinite number of terms of GP is 4 and the sum of their cubes is 192. Find the series.

Evaluate 3. 3

^{1/2}. 3^{1/4}. 3^{1/8}. ................... upto infinity.The sum of first two terms of an infinite G.P. is 5 and each terms is three times the sum of the succeeding terms. Find the G.P.

a man was pointed in the grade of rs.7000-400-15000.in which year of his service will he be drawing salary of rs.10200

if x1,x2,x3 ....xn are in H.P then prove that x1x2+x2x3+x3x4+....+xn-1xn =( n-1)x1xn

the sum of three numbers in GP is 56. if we subtract 1, 7, 21 from these numbers in that order, we obtain an AP. find the numbers.

If one A.M.,

Aand two geometric meansG_{1}_{ }andG_{2}_{ }inserted between any two positive numbers,show thatG._{1}^{2}/ G_{2}+ G_{22}/ G_{1 }= 2Aind the sum of n terms of :- 1/2*5+1/5*8+1/8*11+........

if in a gp the (p+q)th term is a and (p-q)th term is b , prove that the pth term is sqrt (ab).

if sum of 40 A.Ms between two numbers is 120,find the sum 50 AMs between them.

pth term of an A.P. is 'a' and qth term is 'b'. prove that sum of (p+q)th term is p+q/2[a+b+(a-b/p-q)]. plz ans needed soon.....

Let S be the sum, P be the product and R be the sum of reciprocals of n terms in a G.P. Prove that P

^{2}R^{n}= S^{n}Find the sum of 30th term of the sequence 7, 7.7, 7.77, 7.777..................

2. SUM 1 + 3/4+7/16+15/64+31/256+.....TO INFINITY

SUM 1+2/2+3/2^2+4/2^3+.......TO N TERMS

sum of an infinite gp is 57 and sum of their cube is 9747 find the gp

let x=1+a+a^2........ and y=1+b+b^2..... , where modulus(a)

1+ab+a^2b^2.......=xy/(x+y-1)

please i have a test tommorow.....!!

Prove that a,b,c are in A.P. iff 1/bc,1/ca,1/ab are also in A.P.

_{c}a, log_{b}c, log_{a}b are in A.P. then find the common difference of the A.P.1.If the AM ,GM and HM of the first and last terms of the series 25,26,27,.....N-1,N are the terms of the series,find the value of N?

2.The sum of all possible products of the first 'N' natural numbers taken two bu two is:

a)1/24n(n+1)(n-1)(3n+2)

b)n(n+1)(n-1)(2n+3)/24

c)n(n+1)(n-1)(2n+1)/6

d) none of these

If p , q, r are in G.P and the equation px^2 + 2qx + r = 0 and dx^2 + 2ex + f = 0 have a common root , then show that d/p , e/q , f/r are in A.P.

solve the equation 2+5+8...+x=155

The sum of the infinite G.P is 15, Sum of squares of 3 terms of the g.p is 45. Find the series.

the sum of all possible products of first n natural numbers taken two at a time is

ANS:(1/24)*n(n+1)(n-1)(3n+2)

Find the sum of sequence, 7, 77, 777, 7777..... to n terms ???

plz ans fast....

its very urgent

$2.Ifxyz=\left(1-x\right)\left(1-y\right)\left(1-z\right)where0\le x,y,z\le 1,thentheminimumvalueofx\left(1-z\right)+y\left(1-x\right)+z\left(1-y\right)is:\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(a\right)\frac{3}{2}\left(b\right)\frac{1}{4}\left(c\right)\frac{3}{4}\left(d\right)\frac{1}{2}$

5+55+555...n terms. Change this term into G.P. then find the sum of this series?

If (a1,a2,a3......an) are in A>P> with common difference d. then the sum of the series: sin d [cosec a1.cosec a2 + cosec a2.cosec a3 +........+ cosec a(n-1).cosec an ] is what ??

7^2+9^2+11^2+............n terms