Advertisements
Advertisements
Question
Show that 9n+1 – 8n – 9 is divisible by 64, whenever n is a positive integer.
Solution
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
RELATED QUESTIONS
Expand the expression (1– 2x)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 (x + 1)6 + (x – 1)6. Hence or otherwise evaluate `(sqrt2 + 1)^6 + (sqrt2 -1)^6`
Prove that `sum_(r-0)^n 3^r ""^nC_r = 4^n`
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 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.
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.
Show that \[2^{4n + 4} - 15n - 16\] , where n ∈ \[\mathbb{N}\] is divisible by 225.
Determine whether the expansion of `(x^2 - 2/x)^18` will contain a term containing x10?
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 the coefficients of x7 and x8 in `2 + x^n/3` are equal, then n is ______.
If (1 – x + x2)n = a0 + a1 x + a2 x2 + ... + a2n x2n , then a0 + a2 + a4 + ... + a2n equals ______.
Find the coefficient of x15 in the expansion of (x – x2)10.
Find the sixth term of the expansion `(y^(1/2) + x^(1/3))^"n"`, if the binomial coefficient of the third term from the end is 45.
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.
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 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.
Let the coefficients of x–1 and x–3 in the expansion of `(2x^(1/5) - 1/x^(1/5))^15`, x > 0, be m and n respectively. If r is a positive integer such that mn2 = 15Cr, 2r, then the value of r is equal to ______.
The positive integer just greater than (1 + 0.0001)10000 is ______.
