S4w2 FC calculating series

Review -

An explicit formula is one where if I know the formula and I know the order number of the term I want, I can find the value of that term by plugging n into the formula. given the sequence

find the 8th term

Find the explicit formula for the following infinite sequence

as a function of n

How would you find the explicit formula for a geometric series like

A recursive formula is one where you need to find the next term in the sequence. For the following sequence ...3,7,11,15...

find the term after the 15

The recursive formula for an arithmetic series is a_{n}=a_{n-1}+d

where d is the common difference. What do you think the recursive formula would be for a geometric series?

Consider the series

given below. reorder the list so that the last term is after the first, the second to last is after the second, etc.

1. 6
2. 9
3. 4
4. 5
5. 2
6. 1
7. 10
8. 3
9. 7
10. 8

looking above, what do you notice about each pair of low/high terms you created?

How many terms are there in the series

If you pair all the terms as we did before (first from top+first from bottom, second from top+second from bottom) how many term pairs do you have?

You should have noticed above all the term pairs added to the same number - what was that number?

multiply that number by the number of term pairs, and you have found the sum of the series

what did you get?

ok, so that works for the simple case, what about the harder case?

how many terms are there as a function of k and j? Consider before you answer - is this a fence post or fence rail problem?

take your pairs - your first pair is is 4j+4k. your second pair is 4(j+1)+4(k-1)

simplify the second pair.

In fact, all your pairs are going to equal what?

how many pairs are you going to have?

In the above problem, there is a simple function notation inside the sum notation. Any function can work as an arithmetic function.

When writing out f(n) inside summation notation you would use

A teacher earns an annual salary of $45,000 for the first year of her employment. Her annual salary increases by $1,750 each year.

What is the common difference?

A teacher earns an annual salary of $45,000 for the first year of her employment. Her annual salary increases by $1,750 each year.

Calculate the total salary she earns in this employment for the first 10 years.

evaluate the following series by finding the first four terms and adding them together.

give the answer as a solved addition problem (ie: a_1+a_2+a_3+a_4=answer)

There is an equation for a standard geometric series. it looks like this -

where S_k is the sum of the series for any given k, where a is the initial term, r is the common ratio.

given that the series above looks like this

multiply both sides by r

What terms does S_n have that that rS_n does not have?

There is an equation for a standard geometric series. it looks like this -

where S_k is the sum of the series for any given k, where a is the initial term, r is the common ratio.

given that the series above looks like this

multiply both sides by r

What terms does rS_n have that that S_n does not have?

once you have both series

and

you can add ar^k to both sides of the first and a to both sides of the second and you get

and

Whats the relationship between S_n+ar^k and rS_n+a?

Spoiler!

Solve for S_n

That is the proof for the equation for geometric series. You don't need to memorize the proof, but it is a kinda fun one. Any questions about the proof?

Find the first five terms of a geometric sequences if the third term is -135 and the fourth term is 405.

The sum of the first n terms of this sequence is

Find the sum of the first 100 terms in this arithmetic sequence.

The sum of the first n terms of this sequence is

The sum of the first n terms is 477. Find the number of terms. (you will need the quadratic equation.)

I have the general rule of a formula, i can find any term simply by using the term number to calculate value. my general rule is a

I have the general rule of a formula, given any term, I can find either the term after it or before it. My general rule is a