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M.2HS. SL.1: develop sigma notation and use it to write series in equivalent form. For example, write

M.4HS.SL.2 : apply the method of mathematical induction to prove summation formulas. For example, verify that

M.4HS.SL.3:

M.4HS.SL.4: apply infinite geometric series models.

develop intuitively that the sum of an infinite series of positive numbers can converge and derive the formula for the sum of an infinite geometric series.

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Given the sequence:

4, 12, 36, 108,…

Write the equation for the nth term in the sequence.

b. Find the first five partial sums of the sequence.

c. Write the fifth partial sum in sigma notation.

d. Determine whether the infinite sum converges or diverges.

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a_n=a₁(r^n-1)

a_n=4(3^n-1)

s₁=4

s₂=4+12=16

s₃=4+12+36=52

S₄=4+12+36+108=160

s₅= 4+12+36+108+324=484

c.

4, 16, 52, 160, 484, …

The sums keep getting larger, so the infinite sum will diverge.

The first step of the problem involves writing an equation for the nth term of the sequence. By looking at the terms of the sequence, you should notice that it is a geometric sequence where each term is being multiplied by 3. Using the general formula for the nth term of a geometric sequence and substituting 4 in for a sub 1and 3 for r, you get the equation a sub n equals four times 3 to the n-1 power. Next, you need to find the first 5 partial sums of the sequence. Before you do this, you need to find the fifth term of the sequence. Do this by multiplying 108 by 3 to get 324. Then, find the partial sums by adding the terms of the sequence together. Then, you can write the fifth partial sum in a new way, sigma notation. The number below the Greek letter sigma tells you where to start, and the number above sigma tells you where to stop. So this notation tells you that you will be adding the first five terms of the sequence. Finally, you need to determine if the infinite sum of the sequence will converge or diverge. By looking at the partial sums, you can see that they are getting larger and larger. Knowing this, you can see that the sums will continue to grow and approach infinity meaning the infinite sum will diverge.