# For Exercise, determine if the nth term of the sequence defines an arithmetic sequence, a geometric sequence, or neither. If the sequence is arithmetic, find the common difference d. If the sequence is geometric, find the common ratio r. a_{n} = 5 pm sqrt{2n}

Question
Polynomial arithmetic
For Exercise, determine if the nth term of the sequence defines an arithmetic sequence, a geometric sequence, or neither. If the sequence is arithmetic, find the common difference d. If the sequence is geometric, find the common ratio r. $$a_{n} = 5 \pm \sqrt{2n}$$

2020-12-25
Step 1 We know the arithmetic sequence is a sequence whose difference between two consecutive terms is constant. The geometric sequence is a sequence whose ratio of two consecutive terms is constant. The nth term of the sequence is $$a_n = 5 + \sqrt{2n}$$
$$a_{n+1} = 5 + \sqrt(2{n+1})$$ Step 2 Difference between two consecutive terms is $$a_{n+1} - a_n = [5 + \sqrt{2(n+1)}] - (5 + sqrt\{2n})$$
$$a_{n+1} - a_n = \sqrt2$$ which is constant Therefore, nth term defines an arithmetic sequence Common difference is $$d = \sqrt2$$ Step 3 Amns: nth term defines an arithmetic sequence Common difference is $$d = \sqrt2$$

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