# Show that 24n+4-15n-16, where n ∈ N is divisible by 225. - Mathematics

Sum

Show that 2^(4n + 4) - 15n - 16, where n ∈ N is divisible by 225.

#### Solution

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.

Concept: Binomial Theorem for Positive Integral Indices
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Chapter 8: Binomial Theorem - Solved Examples [Page 135]

#### APPEARS IN

NCERT Mathematics Exemplar Class 11
Chapter 8 Binomial Theorem
Solved Examples | Q 10 | Page 135
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