Question

prove the following by using the principal of mathematical induction of all
$\bf \: n € N: 1)1) 1 + 3 + 3 {}^{2} + - - 3n {} - 1 = \binom{3n - 1}{2}$
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Answer 1

given,

1+3+3²+...+3^n-1 = 1/2(3^n-1)

let the given statement be p(n) and then

p( n ) = 1+3+3²+...+3^n-1 = 1/2(3^n-1)

putting n=1 in the given statement,we get

L.h. s= 1

and R.H.S= 1/2(3¹-1)

= 2/2

=1 which is equal to R H.S

Thus, p(1) is true.

let p(k) be true , then

p(k)= 1+3+3²+..+3^k-1=1/2(3^k -1)

adding 3^k to both sides

1+3+3²+...+3^k-1+3^k = 1/2(3^k-1)+3k

= 1/2(3^k+1 - 1)

Answer 2

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