induction and recursion

**zpwnchen** · October 24th 2009, 01:56 PM

give a recursive definaition of the sequence { ${a_n}$ } , n=1,2,3,...
a) $a_n = 1+(-1)^n$
b) $a_n = n(n+1)$
c) $a_n = n^2$

help me how to define them

**proscientia** · October 24th 2009, 02:47 PM

(a)

The sequence alternates between $0$ and $2$ starting with $0.$ The sum of two consecutive terms is $2.$ Hence

$a_1=0,\quad a_{n+1}=2-a_n$

(b)

$a_1=2,\quad a_{n+1}=(n+1)(n+2)=n(n+1)\frac{n+2}n=a_n\frac{n+2} n$

(c)

I leave this one to you.

**zpwnchen** · October 24th 2009, 08:50 PM

c) $a_n = a_n +2(n+1) -1 ,a_1=1$ is that correct?

**CaptainBlack** · October 25th 2009, 12:04 AM

Originally Posted by proscientia

(a)

The sequence alternates between $0$ and $2$ starting with $0.$ The sum of two consecutive terms is $2.$ Hence

$a_1=0,\quad a_{n+1}=2-a_n$

(b)

$a_1=2,\quad a_{n+1}=(n+1)(n+2)=n(n+1)\frac{n+2}n=a_n\frac{n+2} n$

(c)

I leave this one to you.

The idea is to remove all reference to $n$ except in subscripts from the expression. That is $a_n$ should depend on $a_{n-1},\ a_{n-2}, \ ...$ and not explicitly on n

CB

**CaptainBlack** · October 25th 2009, 12:08 AM

Originally Posted by zpwnchen

help me how to define them

See here

CB

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