Advertisements
Advertisements
प्रश्न
Show that \[2^{4n + 4} - 15n - 16\] , where n ∈ \[\mathbb{N}\] is divisible by 225.
Advertisements
उत्तर
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
संबंधित प्रश्न
Expand the expression (1– 2x)5
Expand the expression: `(2/x - x/2)^5`
Expand the expression: (2x – 3)6
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.
Show that 9n+1 – 8n – 9 is divisible by 64, whenever n is a positive integer.
Find the coefficient of x5 in the product (1 + 2x)6 (1 – x)7 using binomial theorem.
Find the value of `(a^2 + sqrt(a^2 - 1))^4 + (a^2 - sqrt(a^2 -1))^4`
Using binomial theorem determine which number is larger (1.2)4000 or 800?
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`
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`.
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 .
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)`
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.
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.
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 ______.
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 ______.
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 ______.
If the coefficients of (2r + 4)th, (r – 2)th terms in the expansion of (1 + x)18 are equal, then r is ______.
