Advertisements
Advertisements
प्रश्न
If a1, a2, a3 and a4 are the coefficient of any four consecutive terms in the expansion of (1 + x)n, prove that `(a_1)/(a_1 + a_2) + (a_3)/(a_3 + a_4) = (2a_2)/(a_2 + a_3)`
Advertisements
उत्तर
Let a1, a2, a3 and a4 be the coefficient of four consecutive terms `"T"_(r + 1), "T"_(r + 2), "T"_(r + 3)` and `"T"_(r + 4)` respectively.
Then a1 = coefficient of Tr+1 = nCr
a2 = coefficient of Tr+2 = nCr+1
a3 = coefficient of Tr+3 = nCr+2
And a4 = coefficient of Tr+4 = nCr+3
Thus `(a_1)/(a_1 + a_2) = (""^n"C"_r)/(""^n"C"_r + ""^n"C"_(r + 1))`
= `(""^n"C"_r)/(""^(n + 1)"C"_(r + 1)` .....`(because ""^n"C"_r + ""^n"C"_(r + 1) = ""^(n + 1)"C"_(r + 1))`
= `(n)/(r(n - r)) xx ((r + 1)(n - r))/(n + 1)`
= `(r + 1)/(n + 1)`
Similarly, `(a_3)/(a_3 + a_4) = (""^n"C"_(r + 2))/(""^n"C"_(r + 2) + ""^n"C"_(r + 3))`
= `(""^n"C"_(r + 2))/(""^(n + 1)"C"_(r + 3))`
= `(r + 3)/(n + 1)`
Hence, L.H.S. = `a_1/(a_1 + a_2) + a_3/(a_3 + a_4)`
= `(r + 1)/(n + 1) + (r + 3)/(n + 1)`
= `(2r + 4)/(n + 1)`
And R.H.S. = `(2a_2)/(a_2 + a_2) + a_3/(a_3 + a_4)`
= `(2(""^n"C"_(r + 1)))/(""^n"C"_(r + 1) + ""^n"C"_(r + 2))`
= `(2(""^n"C"_(r + 1)))/(""^(n + 1)"C"_(r + 2))`
= `2 n/((r + 1)(n - r - 1)) xx ((r + 2)(n - r - 1))/(n + 1)`
= `(2(r + 2))/(n + 1)`
APPEARS IN
संबंधित प्रश्न
Expand the expression: (1– 2x)5
Expand the expression (1– 2x)5
Expand the expression: `(2/x - x/2)^5`
Expand the expression: `(x + 1/x)^6`
Using Binomial Theorem, evaluate the following:
(96)3
Using Binomial Theorem, evaluate of the following:
(102)5
Using binomial theorem, evaluate f the following:
(101)4
Using Binomial Theorem, indicate which number is larger (1.1)10000 or 1000.
Find (a + b)4 – (a – b)4. Hence, evaluate `(sqrt3 + sqrt2)^4 - (sqrt3 - sqrt2)^4`
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 a if the coefficients of x2 and x3 in the expansion of (3 + ax)9 are equal.
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.
Using binomial theorem determine which number is larger (1.2)4000 or 800?
Expand the following (1 – x + x2)4
Find the 4th term from the end in the expansion of `(x^3/2 - 2/x^2)^9`
Determine whether the expansion of `(x^2 - 2/x)^18` will contain a term containing x10?
Find the term independent of x in the expansion of `(sqrt(x)/sqrt(3) + sqrt(3)/(2x^2))^10`.
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 ______.
If z = `sqrt(3)/2 + i^5/2 + sqrt(3)/2 - i^5/2`, then ______.
Find the coefficient of x in the expansion of (1 – 3x + 7x2)(1 – x)16.
Find the coefficient of x15 in the expansion of (x – x2)10.
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
The total number of terms in the expansion of (x + a)100 + (x – a)100 after simplification is ______.
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 number of terms in the expansion of (x + y + z)n ______.
The coefficient of a–6b4 in the expansion of `(1/a - (2b)/3)^10` is ______.
Number of terms in the expansion of (a + b)n where n ∈ N is one less than the power n.
