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

If 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 of integers from 1 to 100 that are divisible by 2 or 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 = ?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?Q3.A person was drawing a monthly salary of Rs 2500 in 11th year of servce and a salary of Rs 2900 in the 19th year. Given that pension is half the salary at retirement time, find his monthly pension if he had put in 25 years of service before retirement. Assume that annual increment is constant.

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

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

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

( answer = 1024 )

^{x}=b^{y}=c^{z},where x,y,z are unequal positive number and a,b,c are in G.P.,then x^{3}+z^{3}is? (Answer is >2y^{3}but how?)If a2, b2, c2, are in AP, then show that a/b+c, b/c+a, c/a+b are in AP.

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.

In an A.P.,s

_{3}=6,s_{6}=3,prove that 2(2n+1)S_{n}_{+4}=(n+4)S_{2n+1}.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

2+5+14+41....

Sum the series: 1+3+7+15+31+............to n terms.

The first term of an AP is 2 and the last term is 50. The sum of all these terms is 442. Find the common difference.

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 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) .

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

Ans given: 25th term

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.100

Sigma 3

^{r}(2-2r)/(r+1)(r+2)r=2

Options : -

(A) 1/2 - {3

^{100}/100(101)}(B) 3/2 - {3

^{101}/101(102)}If

a,b,c,dare in G.P, prove that are in G.P.^{th}mean: n^{th}mean is 11:24, then find 'n'find the sum of integers from 1 to 100 that are divisible by 2 or 5?

if S8/S4=97/81 in a G.P. Prove that r=2/3

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

If

a,b,care in GP, prove thata

^{2}b^{2}c^{2}[ ( 1 / a^{3}) + ( 1 / b^{3}) + ( 1 / c^{3}) ] = a^{3}+ b^{3}+ c^{3}The sum of three numbers in G.P. is 21 and the sum of their squares is 189. Find the numbers.

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

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

1+ab+a2b2+.......=xy/x+y-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}^{1/n}-n (B)greater than or equal to n(n+1)^{1/n}/(n+5) (C)less than n(n+1)^{1/n}- n (D) equal to n(n+1)^{1/n}- n^{b-c})x (Y^{c-a}) x(Z^{a-b})=1_{p+q}=m and t_{p-q}=n. Then find t_{p}.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

Find the sum of all natural numbers lying between 100 and 1000, which are multiples of 5.

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

Find the sum to

nterms of the sequence, 6, 66, 666, 6666…If 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 series 2+11+101+1001+… to n terms.

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

Find the sum of n terms of the sequence

( x + 1/x )

^{2}, ( x^{2}+1 /x^{2})^{2}, (x^{3}+ 1/x^{3})^{2},..........a) 1/2 (root 5) b) root 5 c) 1/2 (root 5 - 1) d) 1/2 (1- root 5 )

let x be AM and y ,z be two GM between any two positive number. prove that y

^{3 }+ z^{3}= 2xyz.FIND FOUR NUMBERS IN AP WHOSE SUM IS 20 &SUM OF WHOSE SQUARES IS 120

what is the proof of formula : s

_{n=}n/2(2a=(n-1)dThe 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.

if AM between two numbers exceeds their GM by 2 and the ratio of two numbers is 4, find the numbersIn 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.

pls answer-- in an AP if a

_{p+1}= 2a_{q+1 }prove that a_{3p+1}= 2a_{p+q+1..}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.

find the sum of n terms of :- 1/2*5+1/5*8+1/8*11+........

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

if 1/a,1/b,1/c are in a.p .prove that a/c=(a-b)/(b-c).

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.

If a constant is multiplied to each term of an A.P., the resulting sequence will also be an A.P.

If a constant is divided from each term of an A.P., the resulting sequence will also be an A.P.

This is written in the study material but in these two cases the common difference is not constant then how these are A.P. PLZ. ANS. FAST...(where summation is from n=1 to n=20)

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.

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

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 }= 2Aif 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).

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

_{k}be the sum of first k terms of an A.P. What must this progression be for the ratio S_{kx }/ S_{x }to be independent of x?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}WHAT IS THE REMAINDER WHEN 30

AND WHAT IS THE REMAINDER WHEN 43^{40}IS DIVIDED BY 17^{86}IS DIVIDED BY 5? IS THERE ANY TRICK TO SOLVE SUCH QUESTIONS? PLEASE,PLEEEEEEEEASE MA'AM/SIR HELP ME OUT OF THIS PROBLEMFind the sum of 30th term of the sequence 7, 7.7, 7.77, 7.777..................

^{1/4}.4^{1/8}.8^{1/16}.16^{1/32}.... upto infinity is equal to :1)1

2)1.5

3)2

4)2.5

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

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

plz ans fast....

its very urgent

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.....!!

Find the sum to n terms of the sequence 1,-1,1,-1,.........,(-1)

^{n+1}Prove that a,b,c are in A.P. iff 1/bc,1/ca,1/ab are also in A.P.

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

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

Find two numbers whose difference is 2 and whose AM exceeds the GM by 1/2.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.

AM between two numbers whose sum is 100 is to the GM is 5:4 find the numbers

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

Which term of the series (28−16𝑖)( 27−14𝑖)( 26−12𝑖) +⋯ is (i) purely real (ii) purely imaginary.

if tan(alpha+theta) = ntan (alpha-theta) then show that : (n+1)sin 2theta=(n-1)sin 2alpha

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

^{2}and sigma n^{3}are in G.P, then find the value of n.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 ??

If G1 is the first of n GMs between two positive numbers a and b , then show that G1 to the power n+1 = a tothe power n*b.