Advertisements
Advertisements
प्रश्न
Show that 9n+1 – 8n – 9 is divisible by 64, whenever n is a positive integer.
Advertisements
उत्तर
We have to prove that 9n + 1 - 8n - 9 = 64k
∴ 9 n+1 - 8n - 9 = (8 + 1)n+ 1 - 8n - 9 [put 9 = 8 + 1]
= \[\ce{[^{n + 1}C_0 8^{n + 1} + ... + ^{n + 1}C_{n-2} 8^3 + ^{n + 1} C_{n - 1} 8^2 + ^{n + 1} C_n 8 + ^{n + 1}C_{n + 1}] - 8n - 9}\]
= \[\ce{^{n + 1} C_0 8 ^{n + 1} + ... + ^{n + 1}C_{n - 2} 8^3 + ^{n + 1}C_{n - 1}8^2 + (n + 1) 8 + 1 - 8n - 9}\]
= \[\ce{^{n + 1}C_0 8 ^{n + 1} + ... + ^{n + 1}C_{n - 2}8^3 + ^{n + 1}C_{n - 1}8^2 + 8n + 8 + 1 - 8n - 9}\]
= \[\ce{^{n + 1}C_0 8^{n + 1} + ... + ^{n + 1}C_{n - 2}8^3 + ^{n + 1}C_{n - 1}8^2}\]
= \[\ce{8^2 [^{n + 1}C_0 8^{n - 1} + ... + ^{n + 1}C_{n - 2} 8 + ^{n + 1}C_{n - 1}]}\]
= 64k [where, k = n + 1C0 8n - 1 + .... + n + 1Cn - 1]
Hence, 9n + 1 - 8n - 9 is divisible by 64, whenever n is a positive integer.
APPEARS IN
संबंधित प्रश्न
Expand the expression: `(x + 1/x)^6`
Using Binomial Theorem, evaluate the following:
(96)3
Using binomial theorem, evaluate the following:
(99)5
Using Binomial Theorem, indicate which number is larger (1.1)10000 or 1000.
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.
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.
Show that \[2^{4n + 4} - 15n - 16\] , where n ∈ \[\mathbb{N}\] is divisible by 225.
Expand the following (1 – x + x2)4
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
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)`
The total number of terms in the expansion of (x + a)51 – (x – a)51 after simplification is ______.
If the coefficients of x7 and x8 in `2 + x^n/3` are equal, then n is ______.
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.
Find the coefficient of x4 in the expansion of (1 + x + x2 + x3)11.
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 ______.
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.
The sum of the last eight coefficients in the expansion of (1 + x)16 is equal to ______.
If the coefficients of (2r + 4)th, (r – 2)th terms in the expansion of (1 + x)18 are equal, then r is ______.
