Advertisements
Advertisements
प्रश्न
If n is a positive integer, prove that \[3^{3n} - 26n - 1\] is divisible by 676.
उत्तर
\[3^{3n} - 26n - 1 = {27}^n - 26n - 1 . . . \left( 1 \right)\]
\[\text{ Now, we have: } \]
\[ {27}^n = (1 + 26 )^n \]
\[\text{ On expanding, we get } \]
\[(1 + 26 )^n = ^{n}{}{C}_0 \times {26}^0 +^{n}{}{C}_1 \times {26}^1 + ^{n}{}{C}_2 \times {26}^2 + ^{n}{}{C}_3 \times {26}^3 +^{n}{}{C}_4 \times {26}^4 + . . . ^{n}{}{C}_n \times {26}^n \]
\[ \Rightarrow {27}^n = 1 + 26n + {26}^2 [^{n}{}{C}_2 + ^{n}{}{C}_3 \times {26}^1 + ^{n}{}{C}_4 \times {26}^2 + . . . ^{n}{}{C}_n \times {26}^{n - 2} ]\]
\[ \Rightarrow {27}^n - 26n - 1 = 676 \times \text{ an integer } \]
\[ {27}^n - 26n - 1 \text{ is divisible by } 676\]
\[\text{ Or, }\]
\[ 3^{3n} - 26n - 1 \text{ is divisible by } 676 \left( \text{ From } (1) \right)\]
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/3 + 1/x)^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`
Show that 9n+1 – 8n – 9 is divisible by 64, whenever n is a positive integer.
Find a if the coefficients of x2 and x3 in the expansion of (3 + ax)9 are equal.
Find an approximation of (0.99)5 using the first three terms of its expansion.
Expand using Binomial Theorem `(1+ x/2 - 2/x)^4, x != 0`
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`
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?
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
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 ______.
The coefficient of xp and xq (p and q are positive integers) in the expansion of (1 + x)p + q are ______.
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 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.
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 ______.
The two successive terms in the expansion of (1 + x)24 whose coefficients are in the ratio 1:4 are ______.
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 ______.
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 ______.
