#### Question

For a given sequence \{a_n\} , a_p+a_q=a_{p+q} where p,q∈N^{*}. Prove a_n=2n when a_1=2.

Question

For a given sequence \{a_n\} , a_p+a_q=a_{p+q} where p,q∈N^{*}. Prove a_n=2n when a_1=2.

Let p=1 and q=n

a_p+a_q=a_{p+q}

becomes

a_1+a_n=a_{n+1}

which is the recursive formulas for an arithmetic sequence.

When a_1=2, using the formula for the general term of an arithmetic sequence

a_n = a_1 + (n-1) d

in which d is the common difference of two consecutive terms of the sequence.

Therefore,

a_n=2+2n-2=2n