Advertisements
Advertisements
प्रश्न
If (1 – x + x2)n = a0 + a1 x + a2 x2 + ... + a2n x2n , then a0 + a2 + a4 + ... + a2n equals ______.
विकल्प
`(3^"n" + 1)/2`
`(3^"n" - 1)/2`
`(1 - 3^"n")/2`
`3^"n" + 1/2`
Advertisements
उत्तर
If (1 – x + x2)n = a0 + a1 x + a2 x2 + ... + a2n x2n , then a0 + a2 + a4 + ... + a2n equals `(3^"n" + 1)/2`.
Explanation:
Putting x = 1 and –1 in
(1 – x + x2)n = a0 + a1 x + a2 x2 + ... + a2n x2n
We get 1 = a0 + a1 + a2 + a3 + ... + a2n ......(1)
And 3n = a0 – a1 + a2 – a3 + ... + a2n ......(2)
Adding (1) and (2), we get
3n + 1 = 2(a0 + a2 + a4 + ... + a2n)
Therefore a0 + a2 + a4 + ... + a2n = `(3^"n" + 1)/2`
APPEARS IN
संबंधित प्रश्न
Expand the expression: (1– 2x)5
Expand the expression (1– 2x)5
Using Binomial Theorem, evaluate the following:
(96)3
Using binomial theorem, evaluate f the following:
(101)4
Using binomial theorem, evaluate the following:
(99)5
Find (a + b)4 – (a – b)4. Hence, evaluate `(sqrt3 + sqrt2)^4 - (sqrt3 - sqrt2)^4`
Find (x + 1)6 + (x – 1)6. Hence or otherwise evaluate `(sqrt2 + 1)^6 + (sqrt2 -1)^6`
Prove that `sum_(r-0)^n 3^r ""^nC_r = 4^n`
Find a if the coefficients of x2 and x3 in the expansion of (3 + ax)9 are equal.
If a and b are distinct integers, prove that a – b is a factor of an – bn, whenever n is a positive integer.
[Hint: write an = (a – b + b)n and expand]
Evaluate `(sqrt3 + sqrt2)^6 - (sqrt3 - sqrt2)^6`
Find the value of `(a^2 + sqrt(a^2 - 1))^4 + (a^2 - sqrt(a^2 -1))^4`
Find an approximation of (0.99)5 using the first three terms of its expansion.
Find the value of (1.01)10 + (1 − 0.01)10 correct to 7 places of decimal.
Show that \[2^{4n + 4} - 15n - 16\] , where n ∈ \[\mathbb{N}\] is divisible by 225.
Expand the following (1 – x + x2)4
Evaluate: `(x^2 - sqrt(1 - x^2))^4 + (x^2 + sqrt(1 - x^2))^4`
Determine whether the expansion of `(x^2 - 2/x)^18` will contain a term containing x10?
If n is a positive integer, find the coefficient of x–1 in the expansion of `(1 + x)^2 (1 + 1/x)^n`
Find the coefficient of x50 after simplifying and collecting the like terms in the expansion of (1 + x)1000 + x(1 + x)999 + x2(1 + x)998 + ... + x1000 .
The total number of terms in the expansion of (x + a)51 – (x – a)51 after simplification is ______.
The number of terms in the expansion of (a + b + c)n, where n ∈ N is ______.
If z = `sqrt(3)/2 + i^5/2 + sqrt(3)/2 - i^5/2`, then ______.
Find the sixth term of the expansion `(y^(1/2) + x^(1/3))^"n"`, if the binomial coefficient of the third term from the end is 45.
If the coefficient of second, third and fourth terms in the expansion of (1 + x)2n are in A.P. Show that 2n2 – 9n + 7 = 0.
In the expansion of (x + a)n if the sum of odd terms is denoted by O and the sum of even term by E. Then prove that 4OE = (x + a)2n – (x – a)2n
If the coefficients of (2r + 4)th, (r – 2)th terms in the expansion of (1 + x)18 are equal, then r is ______.
Let `(5 + 2sqrt(6))^n` = p + f where n∈N and p∈N and 0 < f < 1 then the value of f2 – f + pf – p is ______.
