Arithmetic sequence help

Question

Find a formula for the general term a_n of the sequence assuming the pattern of the first few terms continues.

{ 5/2, 5/8, 5/32, 5/128, 5/512, ⋯}

Assume the first term is a₁

a_n =

Answer 1

This is a geometric sequence. The first term is 5/2 and the common ratio is1/4.

The formula is a(n) = a(1)xr^(n-1)

So in this case it will be a(n) = (5/2)(1/4)^(n-1)

Answer 2

For arithmetic sequences, the general formula is a_n = a₁+(n-1)d, where a₁ is the first term, d is the common difference between any two consecutive terms, and n is a specific term in the sequence.

Geometric sequence formula is a_n= a₁r^(n-1), where a₁ is the first term, r is the common ratio between any two consecutive terms.

Check if it's arithmetic by looking for a common difference:

second term minus first term gives:

a₂ - a₁ = (5/8) - (5/2) = -1.875

third term minus second term:

a₃ - a₂ = (5/32) - (5/8) = -0.46875

The terms of the sequence do not have a common difference between consecutive terms so it's not arithmetic.

Check for the common ratio:

a₂ / a₁ = (5/8) ÷ (5/2) = (5/8) × (2/5) = 1/4

I check other terms just to be sure...

a₃ / a₂ = (5/32) ÷ (5/8) = (5/32) × (8/5) = 1/4

The common ratio, r, is 1/4. We can also notice each term gets multiplied by 1/4 to give the next term in the sequence.

I have given you r and in the problem statement, you were given a₁. Use the geometric sequence formula a_n= a₁r^(n-1) to finish the task.

Answer 3

It is a geometric sewquence with the common ratio of 1/4

a_n = (5/2)(1/4)^n-1

Answer 4

there is no common difference, so it's not an arithmetic sequence

there is a common ratio, r = 1/4 each term is 1/4 of the previous term

an = a1(r^(n-1))

an = 2.5(.25)^(n-1) or (5/2)(1/4)^(n-1)