1 Crore+ students have signed up on EduRev. Have you? Download the App 
{a_{n}} and {b_{n}} be two sequences given by for all n∈N, then a_{1} a_{2} a_{3} … a_{n} is equal to
Since a, b, c are in A.P., therefore
If Z_{r}, r = 1, 2, ...,100 are the roots of then the value of
z_{r} are the roots of z^{101} − 1 =0 , except 1
For the sum of coefficient, put x = 1, to obtain the sum is (1 + 1 ‐ 3)^{2134} = 1
The sum ^{r}C_{r} + ^{r+1}C_{r} + ^{r+2}C_{r} + .... + ^{n}C_{r} (n > r) equals
C(n, r) + c(n ‐1, r) + C(n ‐ 2, r) + . . . + C(r, r)
= ^{n+1}C_{r+1} (applying same rule again and again)
(∵ ^{n}C_{r} + ^{n}C_{r‐1} = ^{n+1}C_{r})
The expansion [x^{2} + (x^{6}  1)^{1/2}]^{5} + [x^{2}  (x^{6}  1)^{1/2}]^{5} is a polynomial of degree
Here last term is of 14 degree.
The general term
The term independent of x, (or the constant term) corresponds to x^{18−3r} being x^{0 }or 18 − 3r= 0 ⇒ r = 6 .
Hence, t_{8} is the greatest term and its value is
∴
For first integral term for r = 3;
The number of irrational terms in the expansion of (2^{1/5} + 3^{1/10})^{55} is
(2^{1/5} + 3^{1/10})^{55}
Total terms = 55 + 1 = 56
Here r = 0, 10, 20, 30, 40, 50
Number of rational terms = 6;
Number of irrational terms = 56 ‐ 6 = 50
The number of terms in the expansion of (2x + 3y− 4z)^{n} is
We have, (2x + 3y − 4z)^{n} = {2x + (3y − 4z)}^{n}
Clearly, the first term in the above expansion gives one term, second term gives two terms, third term gives three terms and so on.
So, Total number of term = 1 +2+3+...+n+(n+1)
In the expansion of the coefficient of x^{−10 }will be
Given expansion is
Since, we have to find coefficient of x^{−10}
∴ −12 + 2r = −10 ⇒ r = 1
Now, then coefficient of x^{−10} is ^{12}C_{1}(a)^{11}(b)^{1} = 12a b
If (1 + ax)^{n} = 1 + 8x + 24x^{2} + ….., then the values of a and n are equal to
The product of middle terms in the expansion of is equal to
it has 12 terms in it’s expansion ,
so there are two middle terms (6th and 7th);
The middle term in the expansion of (1 – 2x + x^{2})^{n} is
Here 2n is even integer, therefore, term will be the middle term.
The sum of the binomial coefficients in the expansion of (x^{−3/4} + ax^{5/4})^{n} lies between 200 and 400 and the term independent of x equals 448. The value of a is
^{23}C_{0} + ^{23}C_{2}+ ^{23}C_{4} + .. + ^{23}C_{22} equals
Given sum = sum of odd terms
27 docs150 tests

27 docs150 tests
