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

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

Uniteasy Admin · Accepted Answer

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,
$$a_n=2+2n-2=2n$$

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