Advertisements
Advertisements
Question
If the coefficients of x7 and x8 in `2 + x^n/3` are equal, then n is ______.
Options
56
55
45
15
Solution
If the coefficients of x7 and x8 in `2 + x^"n"/3` are equal, then n is 55.
Explanation:
Since `"T"_("r" + 1) = ""^"n""C"_"r" "a"^("n" - "r") x^"r"` in expansion of (a + x)n
Therefore, T8 = `""^"n""C"_7 (2)^("n" - 7) (x/3)^7`
= `""^"n""C"_7 (2^("n" - 7))/3^7 x^7`
And T9 = `""^"n""C"_8 (2)^("n" - 8) (x/3)^8`
= `""^"n""C"_8 (2^("n" - 8))/3^8 x^8`
Therefore, `""^"n""C"_7 (2^("n" - 7))/3^7`
= `""^"n""C"_8 (2^("n" - 8))/3^8` ....(Since it is given that coefficient of x7 = coefficient x8)
⇒ `"n"/((7)("n" - 7)) xx (8("n" - 8))/"n" = (2^("n" - 8))/3^8 * 3^7/(2^("n" - 7))`
⇒ `8/("n" - 7) = 1/6`
⇒ n = 55
APPEARS IN
RELATED QUESTIONS
Expand the expression (1– 2x)5
Expand the expression: `(x/3 + 1/x)^5`
Using Binomial Theorem, evaluate the following:
(96)3
Using Binomial Theorem, indicate which number is larger (1.1)10000 or 1000.
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.
Evaluate `(sqrt3 + sqrt2)^6 - (sqrt3 - sqrt2)^6`
Find an approximation of (0.99)5 using the first three terms of its expansion.
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.
Expand the following (1 – x + x2)4
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`
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`
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 .
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 ______.
Find the coefficient of x in the expansion of (1 – 3x + 7x2)(1 – x)16.
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
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 ______.
