Find the sum of the first 10 terms of the Geometric sequence 0.5, 1.5, 4.5, 13.5…
Your basic data is
a = .5
r = 3
Now just use the formula
sum(n) = a(r^n -1)/(r-1)
sum(10) = .5(3^10 - 1)/(3-1)
= .....
Ah, the sum of terms in a geometric sequence. Let's calculate this together, shall we?
First, just to clarify, when you say "Geometric sequence 0.5, 1.5, 4.5, 13.5," are you sure you're not just describing the prices of a cup of coffee at different fancy cafes? Because those prices seem to be going up geometrically every time I walk into one.
Alright, back to math. To find the sum of the first 10 terms of a geometric sequence, we can use the formula:
S = (a * (1 - r^n)) / (1 - r)
Where "S" is the sum of terms, "a" is the first term, "r" is the common ratio, and "n" is the number of terms.
So, in our case, the first term "a" is 0.5, the common ratio "r" is 3 (since each term is multiplied by 3 to get the next term), and the number of terms "n" is 10.
Plugging those values into the formula, we get:
S = (0.5 * (1 - 3^10)) / (1 - 3)
And if you work it out, the sum of the first 10 terms of this geometric sequence is... drumroll, please... -196.5!
Wait, that can't be right. Are we doing math or are we calculating the temperature in Celsius outside?
Let me double-check my math, but I'm pretty sure there's been some kind of mistake here.
To find the sum of the first 10 terms of a geometric sequence, we need to use the formula for the sum of a geometric series.
The formula for the sum of the first n terms of a geometric sequence is:
Sn = a * (1 - r^n) / (1 - r)
Where:
- Sn is the sum of the sequence up to the nth term
- a is the first term of the sequence
- r is the common ratio
In this case:
- The first term (a) is 0.5
- The common ratio (r) is 3 (since each term is obtained by multiplying the previous term by 3)
Now, let's substitute these values into the formula to find the sum of the first 10 terms:
S10 = 0.5 * (1 - 3^10) / (1 - 3)
Simplifying the formula:
S10 = 0.5 * (-59048) / (-2)
S10 = 0.5 * 59048 / 2
S10 = 29524
Therefore, the sum of the first 10 terms of the geometric sequence 0.5, 1.5, 4.5, 13.5... is 29524.
To find the sum of the first 10 terms of a geometric sequence, you can use the formula for the sum of a geometric series. The formula is as follows:
S = a * (1 - r^n) / (1 - r)
where:
S is the sum of the geometric series,
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.
In this case, the first term (a) of the geometric sequence is 0.5, and the common ratio (r) is 3 (since each term is obtained by multiplying the previous term by 3).
Plugging these values into the formula, we get:
S = 0.5 * (1 - 3^10) / (1 - 3)
Now let's calculate:
S = 0.5 * (1 - 59049) / (1 - 3)
= 0.5 * (-59048) / (-2)
= -0.5 * 59048 / 2
= -29524
Therefore, the sum of the first 10 terms of the geometric sequence 0.5, 1.5, 4.5, 13.5... is -29524.