Final answer:
To find how many terms of the series 52 + 48 + 44 + ... add up to 360, we use the sum formula for an arithmetic series and solve the resulting quadratic equation for n. The equation can result in more than one solution, hence the potential for a double answer, but only positive integer solutions are relevant.
Step-by-step explanation:
To determine how many terms of the series 52 + 48 + 44 + ... are needed so that the sum is 360, we can use the formula for the sum of an arithmetic series:
S = n/2 × (2a + (n - 1)d)
Where:
- S is the sum of the series
- n is the number of terms
- a is the first term
- d is the common difference between the terms
For the given series, a = 52, d = -4 (since each term decreases by 4), and S = 360.
Substituting the given values into the formula and simplifying:
360 = n/2 × (104 + (n - 1)×(-4))
Solve for n:
720 = 104n - 4n(n - 1)
720 = 104n - 4n² + 4n
0 = -4n² + 108n - 720
0 = n² - 27n + 180
Solving this quadratic equation, we get the possible values for n. The equation can have more than one solution since it is quadratic, leading to a possible double answer for the number of terms. However, only the positive integer values for n that make sense in the context of the series should be considered.