Advertisements
Advertisements
प्रश्न
Show that `2^(4n + 4) - 15n - 16`, where n ∈ N is divisible by 225.
Advertisements
उत्तर
We have `2^(4n + 4) - 15n - 16`
= `2^(4(n + 1)) - 15n - 16`
= `16^(n + 1) - 15n - 16`
= `(1 + 15)^(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"_3 15^3 + ... + ""^(n + 1)"C"_(n + 1) (15)^(n + 1) - 15n - 16`
= `1 + (n + 1)15 + ""^(n + 1)"C"_2 15^2 + ""^(n + 1)"C"_3 15^3 + ... + ""^(n + 1)"C"_(n + 1) (15)^(n + 1) - 15n - 16`
= `1 + 15n + 15 + ""^(n + 1)"C"_2 15^2 + ""^(n + 1)"C"_3 15^3 + ... + ""^(n + 1)"C"_(n + 1) (15)^(n + 1) - 15n - 16`
= `15^2 [""^(n + 1)"C"_2 + ""^(n + 1)"C"_3 15 + ... "so on"]`
Thus, `2^(4n + 4) - 15n - 16` 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: `(x + 1/x)^6`
Using Binomial Theorem, evaluate of the following:
(102)5
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.
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.
Find the rth term in the expansion of `(x + 1/x)^(2r)`
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`
Find the coefficient of x11 in the expansion of `(x^3 - 2/x^2)^12`
Determine whether the expansion of `(x^2 - 2/x)^18` will contain a term containing x10?
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 ______.
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 ______.
The number of terms in the expansion of (x + y + z)n ______.
The sum of the last eight coefficients in the expansion of (1 + x)16 is equal to ______.
Let `(5 + 2sqrt(6))^n` = p + f where n∈N and p∈N and 0 < f < 1 then the value of f2 – f + pf – p is ______.
The positive integer just greater than (1 + 0.0001)10000 is ______.
