# The general term of a sequence is given a_{n} = n + 5. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference, if it is geometric, find the common ratio.

Question
Polynomial arithmetic
The general term of a sequence is given $$a_{n} = n + 5$$.
Determine whether the sequence is arithmetic, geometric, or neither.
If the sequence is arithmetic, find the common difference, if it is geometric, find the common ratio.

2021-02-10
Step 1
Given: $$a_{n} = n+5$$.
for finding given sequence is arithmetic or geometric we find first three terms then check that given series is A.P. or G.P.
for finding first three term we put $$n = 1,2,3$$ respectively
Step 2
So,
$$a_{1}=1+5=6$$
$$a_{2}=2+5=7$$
$$a_{3}=3+5=8$$
here, first three terms are 6,7 and 8.
now checking for arithmetic progression
we know that in arithmetic progression
2(middle term)=first term+third term
so, putting values and checking
$$2(7)=14$$ and $$6+8=14$$
here, 2(middle term)=first tem + third term
so, given sequence is arithmetic progression
now we can find common difference
we know that common difference of A.P. is given by :(second term)−(first term)
so, common ratio $$=7−6=1$$
hence, common ratio of given series is 1.

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