Question

What is \(S_7\) for 6 – 24 + 96 – 384 + ... ?

Answer 1

I'd suggest you start by determining whether or not this is a geometric series. If it is, then you'll need to identify a, the first term of the sequence, and r, the common ratio. Familiar with those?

Answer 2

It is a geometric series.
\[a_1 = 6\]
\[6 \times -4 = -24 \times -4 = 96\]

Not sure how I find the common ratio tho

Answer 3

The ratio is –4. Just multiply it times each successive terms The individual terms are:

6
-24
96
-384
1536
-6144
24576

Now just add each next term to the previous partial sums S1, S1, .... and get

6 =S1
–18= –24 + 6 = S2
78 = 96 + (–18) = S3 ... etc
–306
1230
–4914
19662 = S7

While this is easily done with a summation formula in one step, I think you need to go through the process manually at least once to understant just what a partial sum is.

Answer 4

i think

Answer 5

r is the common ratio. This means that if you multiply your first term by r, your next term will be -24. As bvb says, the common ratio is -4. It's very important that you be able to determine that your self. What I'd do would be to write 6r=-24 and then solve that for r.

bvb is right in that you can answer this question by finding the 1st 7 terms of the series and then add them together for the 7th partial sum. But I urge you to do some research to find the formula for the nth partial sum of a geometric series.