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
To find the number of terms in a sequence, you can analyze the pattern and determine the relationship between consecutive terms. Let's work through each question step by step.
1. To find the number of terms in the sequence 2, -6, 18, -54, ..., -4374:
The given sequence appears to be a geometric sequence, where each term is multiplied by a common ratio to get the next term.
To find the common ratio, we can divide any term by its preceding term. Let's test it by dividing the third term (-6) by the second term (2):
Common ratio (r) = (-6) / 2 = -3
Now, we can use the formula to find the number of terms (n) in a geometric sequence:
n = log(base r)[(last term / first term) + 1]
Substituting the given values into the formula:
n = log(base -3)[(-4374 / 2) + 1]
Using a logarithmic calculator or software, you can calculate the value of n. In this case, n would be approximately 7.
By calculating the value of n, we find that there are 7 terms in the sequence.
2. To find the common ratio (r) and the value of a1 using the information given for a geometric sequence:
In a geometric sequence, each term is obtained by multiplying the preceding term by a common ratio. We can find the common ratio (r) by dividing any term by the preceding term.
a3 = 324 (third term)
a7 = 64 (seventh term)
Common ratio (r) = a7 / a3 = 64 / 324
To find the value of a1 (first term), we can use the formula:
a1 = a3 / (r^2)
Substituting the values, we have:
a1 = 324 / ((64 / 324)^2)
Evaluating the expression, you can find the value of a1.
3. To find the indicated sum, given a1 = 2, r = -3, and the sum to the 8th term (s8):
The formula to find the sum of the first n terms in a geometric sequence is:
Sn = a1(1 - r^n) / (1 - r)
Substituting the given values, we can find s8:
s8 = 2(1 - (-3)^8) / (1 - (-3))
Calculating this expression will give you the value of s8.
4. To understand the expression represented as an "E" symbol with a 10 on top and k=1 on the bottom, moving to the right, and 2^k as the term:
This represents the series where k starts from 1 and goes up to 10, with each term being equal to 2 raised to the power of k.
The series can be written in summation notation as:
∑(k=1 to 10) 2^k
To calculate the sum, you can plug in the values of k into the term 2^k and add them up:
s = 2^1 + 2^2 + 2^3 + ... + 2^10
Evaluating this expression will give you the sum of the series.