the sum of the first n terms of an arithmetic series is -51. the series has a constant difference of -1.5 and a first term of four. Find the number of terms in the sum
I find using an EXCEL spreadsheet is very helpful for these types of problems. Start with 4 and keep adding 1.5. Then sum the series ending at each of the values. I get n=12
To find the number of terms in the sum:
1. Start with the series' first term, which is given as 4.
2. Determine the constant difference between each term. In this case, it is -1.5.
3. Now, find the sum of the series, which is given as -51.
4. To determine the number of terms, you can use the formula for the sum of an arithmetic series.
The formula for the sum of an arithmetic series is:
Sum = (n/2)(2a + (n-1)d)
Where:
- "Sum" represents the sum of the series (-51 in this case).
- 'n' represents the number of terms in the series (which we are trying to find).
- 'a' represents the first term of the series (4 in this case).
- 'd' represents the common difference between each term (-1.5 in this case).
Substituting the given values into the formula:
-51 = (n/2)(2*4 + (n-1)*(-1.5))
Simplifying the equation:
-51 = (n/2)(8 - 1.5n + 1.5)
-51 = (n/2)(9 - 1.5n)
-51 = 9n/2 - (1.5n²)/2
Multiplying both sides of the equation by 2 to eliminate the fraction:
-102 = 9n - 1.5n²
Rearranging the equation:
1.5n² - 9n - 102 = 0
This is a quadratic equation, and we can solve it using various methods, such as factoring, completing the square, or using the quadratic formula.
To make things easier, we can try factoring the quadratic equation:
1.5n² - 9n - 102 = 0
Dividing the equation by 1.5 to simplify it:
n² - 6n - 68 = 0
Now we need to find two numbers that multiply to -68 and add up to -6 (the coefficient of 'n').
After some trial and error, we find that -2 and -34 satisfy these conditions.
Therefore, the factored form of the equation becomes:
(n + 2)(n - 34) = 0
Now we have two possible solutions:
n + 2 = 0 or n - 34 = 0
Solving for 'n' in each case:
n + 2 = 0
n = -2
n - 34 = 0
n = 34
The two solutions are n = -2 and n = 34. However, in the context of this problem, the number of terms (n) cannot be negative. Therefore, we can discard n = -2 as an extraneous solution.
Hence, the number of terms in the sum is n = 34.