Advertisements
Advertisements
Question
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
Solution
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
RELATED QUESTIONS
Expand the expression: (1– 2x)5
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 of the following:
(102)5
Using binomial theorem, evaluate f the following:
(101)4
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`
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.
Find the coefficient of x5 in the product (1 + 2x)6 (1 – x)7 using binomial theorem.
Find an approximation of (0.99)5 using the first three terms of its expansion.
Expand using Binomial Theorem `(1+ x/2 - 2/x)^4, x != 0`
Using binomial theorem determine which number is larger (1.2)4000 or 800?
Find the value of (1.01)10 + (1 − 0.01)10 correct to 7 places of decimal.
Find the 4th term from the end in the expansion of `(x^3/2 - 2/x^2)^9`
Evaluate: `(x^2 - sqrt(1 - x^2))^4 + (x^2 + sqrt(1 - x^2))^4`
Find the coefficient of x11 in the expansion of `(x^3 - 2/x^2)^12`
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`.
Show that `2^(4n + 4) - 15n - 16`, where n ∈ N is divisible by 225.
If n is a positive integer, find the coefficient of x–1 in the expansion of `(1 + x)^2 (1 + 1/x)^n`
Which of the following is larger? 9950 + 10050 or 10150
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 z = `sqrt(3)/2 + i^5/2 + sqrt(3)/2 - i^5/2`, then ______.
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 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 ______.
