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: `(2/x - x/2)^5`
Expand the expression: `(x/3 + 1/x)^5`
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 (x + 1)6 + (x – 1)6. Hence or otherwise evaluate `(sqrt2 + 1)^6 + (sqrt2 -1)^6`
Show that 9n+1 – 8n – 9 is divisible by 64, whenever n is a positive integer.
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 a if the coefficients of x2 and x3 in the expansion of (3 + ax)9 are equal.
Using binomial theorem determine which number is larger (1.2)4000 or 800?
Find the rth term in the expansion of `(x + 1/x)^(2r)`
Expand the following (1 – x + x2)4
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`
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
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 (1 – x + x2)n = a0 + a1 x + a2 x2 + ... + a2n x2n , then a0 + a2 + a4 + ... + a2n equals ______.
The number of terms in the expansion of (a + b + c)n, where n ∈ N is ______.
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 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.
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 two successive terms in the expansion of (1 + x)24 whose coefficients are in the ratio 1:4 are ______.
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 ______.
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 ______.
