Question

For the following​ sequence, determine if it is an arithmetic​sequence, a geometric​ sequence, or neither. If it is either arithmetic or​ geometric, give the next term in the sequence.
5​, 8​,13​, 21​, 34​, 55,...

Answer 1

Answer: An arithmetic sequence is of the form:

aₙ = a₀ + (n -1)*d

So the difference between a term and the nextone is always the same, and in our case the numbers are 5,8,13,21,34,55.

The distance between the numbers is not constant, so this is not a arithmetic sequence.

a geometric sequence is of the form: aₙ = a*rⁿ

So the therms grow exponentially.

here a₀ = a*r⁰ = a = 5.

a₁ = a*r¹ = 5*r = 13, then r = 13/5 = 2.6

a₂ = a*r² = 5*2.6*2.6 = 33.8, so here we can see that this is not our series, so our series isn't geometric.

So the answer for the first part is neither.

Now, the sequence is  5,8,13,21,34,55, ...

you can see that

5 + 8 = 13

13 + 8 = 21,

21 + 34 = 55

So our sequence is of the form: aₙ = aₙ₋₁ + aₙ₋₂.

then, the next term will be: 55 + 21= 76.