explicit and recursive

Question

find the 7th term in the sequence with the following definition

a₁=64

a_n= a_n-1/2

Answer 1

a_n = (a_n-1) / 2

The common ratio is 1/2.

a₇ = a₁(1/2)⁶

a₇ = 64(1/64)

a₇ = 1

Answer 2

Sometimes these sequence equations are confusing with all the a_n and a_n-1 business but what it means (in this case) is that if you are looking for a particular term (let's call it a_n), then take the one just before that (a_n-1) and divide it be 2.

That makes it a bit tricky if you are looking for the 7^th term because you would need the 6^th term (the one right before the 7^th) so that you can divide that 6^th term by 2. This is called the recursive formula because you have to keep repeating to get to the term you want. So one way to do this problem, although a bit tedious since it requires all this repeating, is to start with the 1st term, divide by 2 and get the second term, so a₂ = 64/2 = 32. Then, continuing, a₃ = 16, a₄ = 8, a₅ = 4, a₆ = 2, and finally, a₇ = 1.

Another way is to figure out the "explicit" equation. The explicit formula has a couple of forms. First, we need to decide if this is an arithmetic sequence or a geometric sequence. Arithmetic means you add/subtract to get to the next term. Geometric means you have to multiply/divide to get to the next term. This is a Geometric Sequence because you must divide by 2 (or multiply by 1/2) to get to the next term.

Method 1: Use the same basic form as an exponential equation A = A₀(r)^x where A is the value, A0 is the value at x = 0, r is the multiplier, and x is the number of periods of terms. If you take this equation and apply it to sequences, you could write a_n = a₀(r)ⁿ where an is the term you are interested in, a₀ is the "zeroth term" [not really a real thing but it's the term right before the first term in you projected backwards], r is the multiplier, and n is the number of the term you are looking for. In this case, a0 would be 128 because I would take the first term and (to go backwards) multiply by 2 to get 128. I'm looking for a₇ so n = 7. And the multiplier (r) is 1/2. So a_n = 128(1/2)⁷. If you plug that into a calculator, you get a₇ = 1.

The second way to writhe the explicit equation is to use a variation of the above that uses the first term instead of the "zeroth term". It goes like this: a_n = a₁(r)^n-1. Although it's more straight forward because you just use the first term and don't need to figure out what the "zeroth term is", it's one more thing to memorize. Anyway, in this case, since a₁ = 64, n = 7, and r = 1/2, it becomes a₇ = 64(1/2)^7-1 or a₇ = 64(1/2)⁶. Again, plugging that into a calculator gets you a₇ = 1