help me because i don't

Question

Enter a recursive rule and an explicit rule for the arithmetic sequence. Then, find the 20th term of the sequence.

25, 35, 45, 55, ...

The recursive rule is f(1) = _____ , f(n) = f (n or n+1 or n-1) +______

The explicit rule is f(n) = ______ +________ (n+1 or n-1 or n)

f(20) = _______

Answer 1

We're given the first four terms of a sequence you're calling f(n): 25,35,45,55.

This is an "arithmetic series" because it changes by a constant amount each term. We call that amount, d, the "common difference". Here, d=10.

We're writing formulas for f(n), so we assign n to each term based on it's order, as in "the n^th term in the sequence".

Call the first term f(1).

f(1)=25. and n=1.

The recursive formula is easy. It just starts with any term and adds the common difference. So f(n), the "nth term" is the term before that, f(n-1), plus 10.

For that f(n-1), if it doesn't make sense, think of what n is, subtract one, then you have the term before it. Example: for the third term, n=3. The term before that is the second term, or (n-1)=(3-1)=2.

So the recursive formula is just f(n) = f(n-1) +10. In words, the (n-1) term is the term before the n^th term plus 10.

We can only use this if we have the term before, so we'll need the *explicit* formula.

-->If you have trouble remembering which type of formula is which, see the word "recur" in recursive, because the recursive formula depends on what came before.

Think of where you've seen the word explicit. "Explicit" means specific, leaving nothing to the imagination. So we need a formula that gives us a specific value without having to use the previous term.

For the explicit formula, we just need the first term, f(1), and the common difference, d.

The general formula looks like this: f(n) = f(1) + d(n-1).

In this example, f(n) = 25 + 10(n-1).

Distribute "d" for an even shorter version: f(n) = 25 + 10n - 10.

f(n) = 15+ 10n.

Use n=20 to get the 20^th term: f(20) = 15 + 10(20) = 15 + 200 = 215.

If this still seems confusing, check out Dr. Khan's video introducing them: https://bit.ly/3i9p9VD

I hope this helps! Let me know if you have any questions...

Thanks,

Liz Z.

Answer 2

d = 10

a_n+1 = a_n + 5

a_n = 25 + (n - 1)(5)

a₂₀ = 25 + (19)(5)