Question

identify the following series as arithmetic, geometric, both, or neither.\\(\sum_{n=1}^{\infty}(3 + na)\\)\
both\
arithmetic\
geometric\
neither

Explanation:

Step1: Recall definitions

An arithmetic series has the form $\sum (a + (n - 1)d)$ (or equivalent, where the difference between consecutive terms is constant). A geometric series has the form $\sum ar^{n - 1}$, where the ratio between consecutive terms is constant.

Step2: Analyze the given series

The general term of the series is $a_n=3 + na$. Let's find the difference between consecutive terms: $a_{n + 1}-a_n=[3+(n + 1)a]-(3+na)=3+na+a - 3 - na=a$. The difference is constant ($a$), so it has the property of an arithmetic series.

Now check for geometric: The ratio $\frac{a_{n+1}}{a_n}=\frac{3+(n + 1)a}{3+na}$. This ratio is not constant (since the numerator and denominator are linear in $n$ and not multiples of each other in a way that the $n$ terms cancel out to give a constant ratio). So it is not geometric. Since the difference is constant, it is arithmetic.

Answer:

arithmetic