Find the number of terms in each sequence.
1. 2,-6, 18,-54,,,,,-4374
Find the common ratio r and the value of a1 using the information given. (assume r >0)
2.a3=324, a7=64
Find the indicated sum.
3. a1=2, r=-3; find s8
4. E like symbol with a 10 on top. On the bottom k=1 to the right 2^k
1. To find the number of terms in a sequence, count the number of terms given. In this case, the sequence is 2, -6, 18, -54, ..., -4374. Counting the terms, we can see that there are 6 terms in this sequence.
2. To find the common ratio (r) and the value of the first term (a1) in a geometric sequence, we need to use the given information. In this case, we are given a3 = 324 and a7 = 64.
We can use the formula for the nth term of a geometric sequence: an = a1 * r^(n-1).
Using a3 = 324, we can substitute n = 3 and find: 324 = a1 * r^(3-1). Simplifying, we get: 324 = a1 * r^2.
Similarly, using a7 = 64, we can substitute n = 7 and find: 64 = a1 * r^(7-1). Simplifying, we get: 64 = a1 * r^6.
Now we have a system of equations:
324 = a1 * r^2
64 = a1 * r^6
To find the common ratio (r) and the value of the first term (a1), we can solve this system of equations simultaneously. One way to approach this is by dividing the second equation by the first equation:
64/324 = (a1 * r^6)/(a1 * r^2)
Simplifying, we get: 2/9 = r^4.
Now we can find the value of r by taking the fourth root of both sides: r = (2/9)^(1/4).
To find a1, we can substitute this value of r back into one of the equations. Let's use the first equation:
324 = a1 * (2/9)^(1/4)^2
Simplifying, we get: 324 = a1 * (2/9)^(1/2).
Now we can solve for a1 by isolating it:
a1 = 324 * (2/9)^(1/2).
Therefore, the common ratio (r) is (2/9)^(1/4) and the value of a1 is 324 * (2/9)^(1/2).
3. To find the sum of an arithmetic series, we can use the formula: Sn = (n/2) * (2a1 + (n-1)d), where Sn is the sum of the first n terms, a1 is the first term, and d is the common difference.
In this case, we are given a1 = 2, r = -3, and we need to find s8 (the sum of the first 8 terms).
Let's plug these values into the formula:
s8 = (8/2) * (2 * 2 + (8-1) * -3).
Simplifying, we get: s8 = 4 * (4 - 21).
s8 = -68.
Therefore, the sum of the first 8 terms is -68.
4. To find the sum represented by the "E" like symbol with a 10 on top and k = 1 on the bottom, where each term is 2^k, we can use the formula for a sum of a geometric series:
S = a * (r^n - 1) / (r - 1)
In this case, a = 2^1 = 2 (the first term), r = 2 (common ratio), and n = 10 (number of terms).
Plugging in these values into the formula, we get:
S = 2 * (2^10 - 1) / (2 - 1)
Simplifying, we have:
S = 2 * (1024 - 1) / 1
S = 2 * 1023 = 2046
Therefore, the sum represented by the "E" like symbol with a 10 on top and k = 1 on the bottom is 2046.