Advertisements
Advertisements
प्रश्न
If n is a positive integer, prove that \[3^{3n} - 26n - 1\] is divisible by 676.
Advertisements
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 + 1/x)^6`
Using Binomial Theorem, evaluate the following:
(96)3
Using Binomial Theorem, evaluate of the following:
(102)5
Using binomial theorem, evaluate the following:
(99)5
Find (x + 1)6 + (x – 1)6. Hence or otherwise evaluate `(sqrt2 + 1)^6 + (sqrt2 -1)^6`
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.
If a and b are distinct integers, prove that a – b is a factor of an – bn, whenever n is a positive integer.
[Hint: write an = (a – b + b)n and expand]
Find the value of (1.01)10 + (1 − 0.01)10 correct to 7 places of decimal.
Show that \[2^{4n + 4} - 15n - 16\] , where n ∈ \[\mathbb{N}\] is divisible by 225.
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`
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 (1 – x + x2)n = a0 + a1 x + a2 x2 + ... + a2n x2n , then a0 + a2 + a4 + ... + a2n equals ______.
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.
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 ______.
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 ______.
The two successive terms in the expansion of (1 + x)24 whose coefficients are in the ratio 1:4 are ______.
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 ______.
The positive integer just greater than (1 + 0.0001)10000 is ______.
