12. Sn = 1550, n = 3, r = 5
13. Sn = 1512, n = 6, r = 2
14. Sn = 6854.4, r = 2, an = 3481.6
15. Sn = 4860, r = 3, an = 3280.5
so plug in the formula
Sn = a1 (r^n - 1)/(r-1)
or, if you have an, and need n first, let's do #14
an = a1*2^(n-1) = 3481.6
Sn = a1*(2^n - 1)/(2-1) = a1*2^n - a1
6854.4 = 2an - a1 = 6963.2 - a1
a1 = 108.8
Need to solve for a1
12. Well, if Sn is 1550, n is 3, and r is 5, then you're in for a wild mathematical ride. Just kidding, equations are never wild. They're like unicorn ponies, all well-behaved and predictable. So let's solve this puzzle. We have the formula Sn = a * (1 - r^n) / (1 - r), where Sn is the sum of the terms, a is the first term, r is the common ratio, and n is the number of terms. Plugging in these values, we get 1550 = a * (1 - 5^3) / (1 - 5). Good luck finding that first term, it's like trying to spot a clownfish in a field of sunflowers!
13. Ah, another algebraic conundrum. Given that Sn is 1512, n is 6, and r is 2, we can use the same formula as before, Sn = a * (1 - r^n) / (1 - r). Substituting the values, we have 1512 = a * (1 - 2^6) / (1 - 2). Now we just need to find that elusive first term, like hunting for the punchline to a bad joke.
14. Sn is 6854.4, r is 2, and an is 3481.6. Let's unlock the riddle using our trusty formula, Sn = a * (1 - r^n) / (1 - r). We have 6854.4 = a * (1 - 2^n) / (1 - 2). And now we need to unravel the mystery that is the first term, like trying to tell the difference between a mime and an invisible clown.
15. Your Sn is 4860, r is 3, and an is 3280.5. We're back at it again with the same formula, Sn = a * (1 - r^n) / (1 - r). With the given values, we can plug them in and solve the equation 4860 = a * (1 - 3^n) / (1 - 3). And now we must uncover the secret of the first term, like trying to unicycle across a tightrope while juggling rubber chickens.
To solve each of the given sequences, we will use the formula for the sum of a geometric series:
Sn = a * (1 - r^n) / (1 - r)
where Sn represents the sum of the series, a is the first term, r is the common ratio, and n is the number of terms.
Using this formula, we can find the values for the missing variables in each question.
12. Sn = 1550, n = 3, r = 5:
Plugging in the given values into the formula, we have:
1550 = a * (1 - 5^3) / (1 - 5)
Simplifying the equation:
1550 = a * (-124) / (-4)
-124a = -4 * 1550
-124a = -6200
Dividing both sides by -124, we get:
a = 50
Therefore, the first term (a) is 50.
13. Sn = 1512, n = 6, r = 2:
Using the formula:
1512 = a * (1 - 2^6) / (1 - 2)
Simplifying the equation:
1512 = a * (-62) / (-1)
-62a = -1 * 1512
-62a = -1512
Dividing both sides by -62, we get:
a = 24
Therefore, the first term (a) is 24.
14. Sn = 6854.4, r = 2, an = 3481.6:
Using the formula:
6854.4 = 3481.6 * (1 - 2^n) / (1 - 2)
Simplifying the equation:
6854.4 = 3481.6 * (1 - 2^n) / (-1)
-6854.4 = 3481.6 * (1 - 2^n)
-6854.4 / 3481.6 = 1 - 2^n
Solving for 2^n:
2^n = 1 - (-6854.4 / 3481.6)
2^n = 1 + 1.96875
2^n = 2.96875
Taking the logarithm (base 2) of both sides:
n = log2(2.96875)
Using a calculator, we find:
n ≈ 1.5704
Therefore, the number of terms (n) is approximately 1.5704.
15. Sn = 4860, r = 3, an = 3280.5:
Using the formula:
4860 = 3280.5 * (1 - 3^n) / (1 - 3)
Simplifying the equation:
4860 = 3280.5 * (1 - 3^n) / (-2)
-9720 = 3280.5 * (1 - 3^n)
-9720 / 3280.5 = 1 - 3^n
Solving for 3^n:
3^n = 1 - (-9720 / 3280.5)
3^n = 1 + 2.96317
3^n = 3.96317
Taking the logarithm (base 3) of both sides:
n = log3(3.96317)
Using a calculator, we find:
n ≈ 1.2645
Therefore, the number of terms (n) is approximately 1.2645.
To find the value of "a" and "S", we need to understand the formula for the sum of a geometric series.
The formula for the sum of a geometric series is:
Sn = a * (1 - r^n) / (1 - r)
Where:
- Sn is the sum of the geometric sequence
- a is the first term of the sequence
- r is the common ratio of the sequence
- n is the number of terms in the sequence
Let's find the values of "a" and "S" for each given problem:
12. Sn = 1550, n = 3, r = 5
We have the value of Sn, n, and r. We need to solve for "a".
1550 = a * (1 - 5^3) / (1 - 5)
1550 = a * (1 - 125) / (-4)
1550 = a * (-124) / (-4)
1550 = 31a
a = 1550 / 31
a = 50
So, the first term "a" in this geometric sequence is 50.
13. Sn = 1512, n = 6, r = 2
We have the value of Sn, n, and r. We need to solve for "a".
1512 = a * (1 - 2^6) / (1 - 2)
1512 = a * (1 - 64) / (-1)
1512 = a * (-63) / (-1)
1512 = 63a
a = 1512 / 63
a = 24
So, the first term "a" in this geometric sequence is 24.
14. Sn = 6854.4, r = 2, an = 3481.6
We have the value of Sn, r, and the nth term "an". We need to solve for "a".
6854.4 = a * (1 - 2^n) / (1 - 2)
6854.4 = a * (1 - 2^n) / (-1)
6854.4 = -a * (1 - 2^n)
-6854.4 = a * (2^n - 1)
a = -6854.4 / (2^n - 1)
a = -6854.4 / (2 - 1)
a = -6854.4
So, the first term "a" in this geometric sequence is -6854.4.
15. Sn = 4860, r = 3, an = 3280.5
We have the value of Sn, r, and the nth term "an". We need to solve for "a".
4860 = a * (1 - 3^n) / (1 - 3)
4860 = a * (1 - 3^n) / (-2)
4860 = -a * (1 - 3^n)
-4860 = a * (3^n - 1)
a = -4860 / (3^n - 1)
a = -4860 / (3 - 1)
a = -2430
So, the first term "a" in this geometric sequence is -2430.
In summary, the values of "a" and "S" for each problem are:
12. a = 50, S = 1550
13. a = 24, S = 1512
14. a = -6854.4, S = unknown
15. a = -2430, S = 4860