Find each of the sums indicated or described. Show use of the appropriate formula, rather than simply using a calculator to add them.
1) The sum of the first 17 terms of a geometric series with a first term of 3000 and a common ratio of 1/2.
2) 7+9+11+13+... 10 terms
3) 1 + �ã2 + 2 + 2�ã2... 12 terms
The "ã" are suppose to be square root symbols.
#1: S = 3000(1-1/2^17)/(1 - 1/2)
#2: S = 10/2 (2*7+2)
#3: S = 1(√2^12 - 1)/(√2-1)
1) The sum of the first 17 terms of a geometric series with a first term of 3000 and a common ratio of 1/2.
To find the sum of a geometric series, we can use the formula:
Sn = a * (1 - r^n) / (1 - r)
Where:
- Sn is the sum of the first n terms
- a is the first term of the series
- r is the common ratio
- n is the number of terms
In this case, we have:
- a = 3000 (the first term)
- r = 1/2 (the common ratio)
- n = 17 (the number of terms)
Plugging these values into the formula, we get:
Sn = 3000 * (1 - (1/2)^17) / (1 - 1/2)
Simplifying the expression, you can calculate the sum of the first 17 terms of the geometric series.
2) 7+9+11+13+... 10 terms
To find the sum of an arithmetic series, we can use the formula:
Sn = (n/2) * (2a + (n-1)d)
Where:
- Sn is the sum of the first n terms
- a is the first term of the series
- d is the common difference between consecutive terms
- n is the number of terms
In this case, we have:
- a = 7 (the first term)
- d = 2 (the common difference)
- n = 10 (the number of terms)
Plugging these values into the formula, we get:
Sn = (10/2) * (2*7 + (10-1)*2)
Simplifying the expression, you can calculate the sum of the first 10 terms of the arithmetic series.
3) 1 + �ã2 + 2 + 2�ã2... 12 terms
It appears that this series alternates between two terms: 1 and 2√2. To find the sum of this sequence, we can add up all the terms.
1 + �ã2 + 2 + 2�ã2 + ...
To find the sum, we can group the terms into pairs:
(1 + �ã2) + (2 + 2�ã2) + ...
Using this grouping, we can also notice that the sum of each pair is equal to 3 + 3�ã2.
Therefore, we can multiply this sum by the number of pairs (which is half the number of terms since each pair has two terms) to find the total sum:
Total sum = (3 + 3�ã2) * (12 / 2)
Simplifying the expression, you can calculate the sum of the 12 terms in the given series.