Anyone please please solve this question......I will mark you as the brainlest....plzzzzzzz........
prove that :
(3)^2n+1 + (2)^n+2 is divisible by 7
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For n = 1, we get 2^3 + 3^3 = 8 + 27 = 35; divisible by 7.
let n = m be divisible by 7. This means 2^(m+2) + 3^(2m+1) = 7·N, where N is a positive integer.
If this is true, what can we conclude about n = m+1?
2^(m+3) + 3^(2m+3) = 2·2^(m+2) + 9·3^(2m+1) = 2·[2^(m+2) + 2·3^(2m+1)] + 7·3^(2m+1) = 7·N + 7·3^(2m+1); this is also divisible by 7.
Therefore, the formula works for n = 1; and if the formula works for n = m, then it also works for n = m+1.
This constitutes the proof by induction.
let n = m be divisible by 7. This means 2^(m+2) + 3^(2m+1) = 7·N, where N is a positive integer.
If this is true, what can we conclude about n = m+1?
2^(m+3) + 3^(2m+3) = 2·2^(m+2) + 9·3^(2m+1) = 2·[2^(m+2) + 2·3^(2m+1)] + 7·3^(2m+1) = 7·N + 7·3^(2m+1); this is also divisible by 7.
Therefore, the formula works for n = 1; and if the formula works for n = m, then it also works for n = m+1.
This constitutes the proof by induction.
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