IIT JEE , AIEEE Forum - Mathematics , Algebra

shailesh_45

1 ] There are twos, threes and five each being 8 in number. Find the

sum of the products of these numbers taking at least one 2,3, and 5.

2] The first term of an infinitly decreasing GP is unity and its sum is S.

Find the sum of the squares of the terms of the progression.

3 ] An AP and an HP have the same first term, the same last term and the same number of terms. Prove that the product of the rth term from the beginning of the first sequence and the rth term from the end of the second sequence is independent of r.

4 ] Find the sum of the product of the integers 1,2,3........, n taken two at a time and show that it is equal to half the excess of the sum of the cubes of the given integers over the sum of their squares.

sboosy

1,r,r^2,.... be the terms
S = 1/(1-r) which means r = 1-1/S .......(1)

after squaring the terms are 1,r^2,r^4,.....
S' = new sum = 1/(1-r^2)
Using 1

we get
S' = new sum = S^2/(2S-1)

ashish_banga

answer to 2 = S^2/ 2S - 1

r = S - 1 / S
and sum 1 + r^2 + r^4 ...........
= 1 / 1 - r^2
put the value of r in this

mukundmadhav

4th question
Consider the expression (1+2+3+4+....n)^2
This will have two kinds of terms,
Squares of individual terms and mixed terms.. The Sum of mixed terms is what you require...
i.e. 1.2 + 2.3

For (a+b+c)^2
it is a^2 +b^2 +c^2 + 2ab +2ac +2bc
Similarly for n terms
[n(n+1)/2]^2= _{[ 1]}

^{[ n]} r^2 + 2(product of terms taken two at a time)
Shift sides and you get the answer

mukundmadhav

3rd question is easy..
Let first term be a, last term be b, number of terms be n
Find common differences in terms of a, b and n
Then substitute the r th and n-r th terms in terms of these variables and you get the answer..

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