can you help me?

Here is what you see this week as an arithmetic sequence, but we are going to look at it in a simpler form.:

2, 4, 6, 8, 10,... not too difficult to figure out _what comes next! This is adding 2 each time, but it's also doubling the position in line! What's the formula? If we call the position in line n, then we can say the pattern is f(n) = 2n. The 25th number is 2*25 = 50.

Often it's important to find the sum of these numbers. When a sequence is arithmetic (adding the same thing each time) we can find a partial sum using the formula:

Sn=n(a1+an)2

where S_nis called the partial sum, n is the position of the last number we are adding, a₁is the first number in the sequence and a_n is the last number we are adding.

Here is what you see this week as what we call a geometric sequence.

800, 200, 50, 12.5, 3.125,... what's next? Is it dividing by 4 or is it powers of 14? Well, a formula might be800*(14)n1 where n is the position in line. We can add the terms in this sequence also. There are two forms of a sum for geometric sequences, one for if that multiplier is greater than 1 and one for if it's between 0 and 1 (exclusive of 0 and 1). We are going to look at both.

For adding the first n numbers, we have the formula

Sn=an(1rn)1r

where a_n is the last number in the sequence to be added and r is what we are multiplying by each time.

For 0 < r < 1, we can use the formula above, but we have the formula

S=a11r

where a₁ is the first number in the sequence and r is the number we are multiplying by each time. This is the infinite sum, or all the numbers in the sequence forever and ever. ("How can that be?" I hear you all ask. Well, think about fractions. As we multiply by a proper positive fraction things get smaller. When we talk about doing that an infinite number of times, it ends up that the terms become 0. Okay, it's more complicated than that and if you really want to know, take Calculus!)

• Post 1 : You will need to post before you can see anyone else's posts. You are going to find the sum of the first 20 terms for each sequence above. You will find the infinite sum for the geometric sequence. In your post, you must show the formula and calculation steps. Post which only have the answers will not receive credit.
• Post 2 : Find an application of either type of sum and post that to the discussion, explaining why it is like one of these sums. You can certainly do a web search! Create problem for another person to solve, providing the necessary information for the formula.