Arithmitic and Geometric sequences

Question

The first three terms of a sequence are given. Round to the nearest thousandth (if necessary).

4,6,9,...

Find the 7th term.

Answer 1

Hello, Mariah,

I see a pattern in the first three digits: Each succeeding digit is more than the preceding one by the step in the sequence. Given that step 1 =. 4, we can see that step 2 can be obtained if we take the vales of the preceding step and add the step number to it. Step 2 would therefore become Step 1 plus the current step number (2, in this case).

Step 2 = Step1 + Current step number (2)

Step2 = 4 + 2

Step2 = 6

Step3 = Step2 + 3

Step3 = 6 + 3, or 9

And so forth.

With no other information, it would be reasonable to assume that this pattern continues:, which would produce the following list:

Bob

Answer 2

The pattern looks as follows

a_n = a_n-1 + n

a₂ = a₁ + 2 = 4 + 2 = 6

a₃ = a₂ + 3 = a₁ + 2 + 3 = 4 +2 + 3 = 9

a_n = 4 + ∑i -1 = 3 + n(n+1)/2

a₁ = 3 + 1 = 4

a₂ = 3 + 2(3)/2 = 6

a₃ = 3 + 3(4)/2 = 9

:

:

a₇ = 3 + 7(8)/2 = 31



Answer 3

6/4 = 1.5

9/6 = 1.5

That means this is a geometric sequence with a common ratio of 1.5

You can use the following formula

an=a1(r)^n-1

a7=4(1.5)^7-1 = 45.5625