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 the sequence, we can look for a pattern. Starting with the first term, 2, each subsequent term is obtained by multiplying the previous term by -3. So the common ratio is -3.
To find the number of terms, we can start with the first term and keep multiplying it by -3 until we reach the last term, -4374.
2 × (-3) = -6
-6 × (-3) = 18
18 × (-3) = -54
-54 × (-3) = 162
162 × (-3) = -486
-486 × (-3) = 1458
1458 × (-3) = -4374
Each step is multiplying by -3, so we can see that there are 8 terms in the sequence.
2. We are given the values of a3 and a7.
In this case, a3 refers to the third term in the sequence, and a7 refers to the seventh term in the sequence.
a3 = 324
a7 = 64
To find the common ratio, we can divide the seventh term by the third term:
common ratio = a7 / a3 = 64 / 324 = 4 / 27
To find the first term (a1), we can use the formula for finding a term in a geometric sequence:
an = a1 * r^(n-1)
For the third term (a3): 324 = a1 * (4/27)^(3-1)
324 = a1 * (4/27)^2
324 = a1 * (16/729)
a1 = 324 * (729/16)
a1 = 1458
Therefore, the common ratio (r) is 4/27 and the first term (a1) is 1458.
3. We are given a1 = 2, r = -3, and we need to find s8.
The formula for the sum of a geometric series is:
Sn = a1 * (1 - r^n) / (1 - r)
Using the given values:
s8 = 2 * (1 - (-3)^8) / (1 - (-3))
s8 = 2 * (1 - 6561) / (1 + 3)
s8 = 2 * (-6560) / 4
s8 = -3280
Therefore, s8 is equal to -3280.
4. The sum indicated is the sum of a series where k ranges from 1 to 10 and each term is 2^k.
The notation can be rewritten as:
∑ (k=1 to 10) 2^k
To calculate this sum, we can substitute each value of k into the expression 2^k and add them all together.
∑ (k=1 to 10) 2^k = 2^1 + 2^2 + 2^3 + ... + 2^10
Using the property of geometric sequences, we can rewrite this sum as:
∑ (k=1 to 10) 2^k = 2 * (1 - 2^10) / (1 - 2)
∑ (k=1 to 10) 2^k = 2 * (1 - 1024) / (-1)
∑ (k=1 to 10) 2^k = 2 * (-1023) / (-1)
∑ (k=1 to 10) 2^k = 2046
Therefore, the indicated sum is equal to 2046.
1. To find the number of terms in the sequence 2, -6, 18, -54,..., -4374, we need to identify the pattern. Notice that each term is obtained by multiplying the previous term by -3. Starting with the first term, we have:
a1 = 2
a2 = 2 * (-3) = -6
a3 = -6 * (-3) = 18
a4 = 18 * (-3) = -54
...
We can see that the common ratio, r, is -3. So, to find the number of terms, we can set up the equation:
a_n = a1 * r^(n-1)
where a_n is the nth term and n is the number of terms. We can substitute the last term (-4374) for a_n and solve for n:
-4374 = 2 * (-3)^(n-1)
Divide both sides by 2 to isolate the exponential term:
-2187 = (-3)^(n-1)
Since -3 to any power will always be negative, there is no solution that satisfies the given equation. Therefore, the sequence does not have an end value.
2. To find the common ratio, r, and the value of a1 in the sequence where a3 = 324 and a7 = 64, we can use the formula for the nth term of a geometric sequence:
a_n = a1 * r^(n-1)
Using the given information, we can substitute the values:
a3 = a1 * r^2 = 324
a7 = a1 * r^6 = 64
Now, we can divide these two equations to eliminate a1:
(a1 * r^2) / (a1 * r^6) = 324 / 64
Simplifying, we get:
r^4 = 324 / 64
r^4 = 5.0625
Taking the fourth root of both sides, we find:
r = ∛(5.0625) ≈ 1.5874
Now we can substitute this value of r into one of the equations to find a1:
a3 = a1 * r^2 = 324
a1 * (1.5874)^2 = 324
a1 * 2.5244 ≈ 324
a1 ≈ 324 / 2.5244
a1 ≈ 128.235
Therefore, the common ratio, r, is approximately 1.5874 and the value of a1 is approximately 128.235.
3. To find the sum, s8, with a geometric sequence defined by a1 = 2 and r = -3, we can use the formula for the sum of the first n terms of a geometric sequence:
s_n = a1 * (1 - r^n) / (1 - r)
Substituting the given values:
s8 = 2 * (1 - (-3)^8) / (1 - (-3))
Simplifying the expression:
s8 = 2 * (1 - 6561) / 4
s8 = 2 * (-6560) / 4
s8 = -6560
Therefore, the sum of the first 8 terms in the sequence is -6560.
4. The symbol you described "E" with a 10 on top and the bottom "k=1" to the right "2^k" represents the summation notation. Specifically, it represents the sum of a series where k starts from 1 and goes up to 10, with each term being 2 raised to the power of k.
So, the expression can be written as:
∑(k=1 to 10) 2^k
To find the value of this sum, we can calculate each term and add them together:
2^1 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8 + 2^9 + 2^10
Using the properties of exponents, we can simplify each term:
2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 + 1024
Adding these terms together:
= 2046
Therefore, the sum of the series represented by the given notation is 2046.