Advertisements
Advertisements
Question
Show that \[2^{4n + 4} - 15n - 16\] , where n ∈ \[\mathbb{N}\] is divisible by 225.
Advertisements
Solution
We have,
\[2^{4n + 4} - 15n - 16 = 2^{4\left( n + 1 \right)} - 15n - 16\]
\[ = {16}^{n + 1} - 15n - 16\]
\[ = \left( 1 + 15 \right)^{n + 1} - 15n - 16\]
\[ =^{n + 1} C_0 {15}^0 +^{n + 1} C_1 {15}^1 +^{n + 1} C_2 {15}^2 + . . . +^{n + 1} C_{n + 1} {15}^{n + 1} - 15n - 16\]
\[ = 1 + (n + 1)15 +^{n + 1} C_2 {15}^2 + . . . +^{n + 1} C_{n + 1} {15}^{n + 1} - 15n - 16\]
\[ = 1 + 15n + 15 +^{n + 1} C_2 {15}^2 + . . . +^{n + 1} C_{n + 1} {15}^{n + 1} - 15n - 16\]
\[ =^{n + 1} C_2 {15}^2 + . . . +^{n + 1} C_{n + 1} {15}^{n + 1} \]
\[ = {15}^2 \left( {}^{n + 1} C_2 + . . . +^{n + 1} C_{n + 1} {15}^{n - 1} \right)\]
\[ = 225\left( {}^{n + 1} C_2 + . . . +^{n + 1} C_{n + 1} {15}^{n - 1} \right)\]
Thus,
\[2^{4n + 4} - 15n - 16\] , where n ∈ \[\mathbb{N}\] is divisible by 225.
APPEARS IN
RELATED QUESTIONS
Expand the expression: (1– 2x)5
Expand the expression (1– 2x)5
Expand the expression: (2x – 3)6
Using Binomial Theorem, evaluate the following:
(96)3
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`
Evaluate `(sqrt3 + sqrt2)^6 - (sqrt3 - sqrt2)^6`
Expand using Binomial Theorem `(1+ x/2 - 2/x)^4, x != 0`
If n is a positive integer, prove that \[3^{3n} - 26n - 1\] is divisible by 676.
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`
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`
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`
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 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 ______.
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.
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
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 ______.
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 sum of the last eight coefficients in the expansion of (1 + x)16 is equal to ______.
