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, we need to first identify the pattern. In this sequence, the terms are multiplying by -3 from one term to the next. Starting with the first term 2, each subsequent term is obtained by multiplying the previous term by -3. So, the sequence is a geometric sequence with a common ratio of -3.
To find the number of terms, we can count the terms in the sequence until we reach the desired term or notice a pattern. Starting with 2, the terms are -6, 18, -54, and so on. We can continue multiplying by -3 until we reach the last term, which is -4374.
So, the number of terms in this sequence is 5.
2. In this problem, we are given two terms (a3 and a7) and need to find the common ratio (r) and the first term (a1).
The common ratio (r) in a geometric sequence is found by dividing any term by its previous term. In this case, a7 is given as 64, and a3 is given as 324. So, we can write:
r = a7 / a3
r = 64 / 324
Simplifying, we get:
r = 2 / 9
To find the value of the first term (a1), we can use the formula for the general term of a geometric sequence:
an = a1 * r^(n-1)
Substituting the values of a3 and r into the equation, we have:
a3 = a1 * (2/9)^(3-1)
324 = a1 * (2/9)^2
Now, we can solve this equation to find the value of a1.
3. To find the sum (s8) of a geometric series, we can use the formula:
s8 = a1 * (1 - r^8) / (1 - r)
In this problem, we are given a1 as 2 and r as -3.
Substituting these values into the formula, we have:
s8 = 2 * (1 - (-3)^8) / (1 - (-3))
Simplifying the equation, we get:
s8 = 2 * (1 - 6561) / (1 + 3)
Now, we can calculate the value of s8.
4. The given symbol is the summation symbol (∑), and it indicates a series. The expression on the right side of the summation symbol is the term of the series, and the index variable (k) shows the range of values.
In this case, the expression is 2^k, and the index variable is k, which ranges from 1 to 10.
So, the series can be written as:
∑ (2^k) (where k ranges from 1 to 10)
To find the value of this sum, we can calculate the individual terms of the series and add them up.
The terms of the series are 2^k, where k ranges from 1 to 10. We can substitute these values into the expression and add them up:
2^1 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8 + 2^9 + 2^10
Now, we can simplify this expression to find the sum.