Advertisements
Advertisements
प्रश्न
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 O2 – E2 = (x2 – a2)n
Advertisements
उत्तर
Given expression is (x + a)n
(x + a)n = nC0xn a0 + nC1xn–1a + nC2xn–2a2 + nC3xn–3a3 + … + nCnan
Sum of odd terms,
O = `""^n"C"_0 x^n + ""^n"C"_2 x^(n - 2)a^2 + ""^n"C"+4x^(n - 4)a^4` + ...
And the sum of even terms,
E = `""^n"C"_1x^(n - 1) * a + ""^n"C"_3x^(n - 3)a^3 + ""^n"C"_5x^(n - 5)a^5` + ...
Now (x + a)n = O + E ......(i)
Similarly (x – a)n = O – E .....(ii)
Multiplying equation (i) and equation (ii), we get,
(x + a)n (x – a)n = (O + E)(O – E)
⇒ (x2 – a2)n = O2 – E2
Hence O2 – E2 = (x2 – a2)n
APPEARS IN
संबंधित प्रश्न
Expand the expression (1– 2x)5
Expand the expression: `(2/x - x/2)^5`
Expand the expression: `(x/3 + 1/x)^5`
Using binomial theorem, evaluate the following:
(99)5
Show that 9n+1 – 8n – 9 is divisible by 64, whenever n is a positive integer.
Prove that `sum_(r-0)^n 3^r ""^nC_r = 4^n`
Find a, b and n in the expansion of (a + b)n if the first three terms of the expansion are 729, 7290 and 30375, respectively.
Find the coefficient of x5 in the product (1 + 2x)6 (1 – x)7 using binomial theorem.
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 the expansion of (3x2 – 2ax + 3a2)3 using binomial theorem.
If n is a positive integer, prove that \[3^{3n} - 26n - 1\] is divisible by 676.
Using binomial theorem determine which number is larger (1.2)4000 or 800?
Find the rth term in the expansion of `(x + 1/x)^(2r)`
Find the 4th term from the end in the expansion of `(x^3/2 - 2/x^2)^9`
Find the coefficient of x11 in the expansion of `(x^3 - 2/x^2)^12`
Find the term independent of x in the expansion of `(sqrt(x)/sqrt(3) + sqrt(3)/(2x^2))^10`.
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 ______.
If (1 – x + x2)n = a0 + a1 x + a2 x2 + ... + a2n x2n , then a0 + a2 + a4 + ... + a2n equals ______.
The coefficient of xp and xq (p and q are positive integers) in the expansion of (1 + x)p + q are ______.
The number of terms in the expansion of (a + b + c)n, where n ∈ N is ______.
Find the coefficient of x in the expansion of (1 – 3x + 7x2)(1 – x)16.
Given the integers r > 1, n > 2, and coefficients of (3r)th and (r + 2)nd terms in the binomial expansion of (1 + x)2n are equal, then ______.
The two successive terms in the expansion of (1 + x)24 whose coefficients are in the ratio 1:4 are ______.
The number of terms in the expansion of (x + y + z)n ______.
Number of terms in the expansion of (a + b)n where n ∈ N is one less than the power n.
The sum of the last eight coefficients in the expansion of (1 + x)16 is equal to ______.
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 ______.
The positive integer just greater than (1 + 0.0001)10000 is ______.
