Final answer:
The arithmetic series with a common difference of 7, starting at 4 and ending at 249, contains 36 terms. To find the sum of this series, we use the formula for the sum of an arithmetic series to obtain the result of 4554.
Step-by-step explanation:
The question involves finding the number of terms in an arithmetic series and then computing the sum of the series.
Number of Terms (a)
The series given is 4 + 11 + 18 + 25 + ... + 249. This is an arithmetic sequence with a common difference of 7 (since 11 - 4 = 7, and so on). To find the number of terms, n, we use the formula for the nth term of an arithmetic series
an = a1 + (n - 1)d, where an is the nth term, a1 is the first term, and d is the common difference. Plugging in the values, we get 249 = 4 + (n - 1)(7).
Solving for n, we get n = 36. So, there are 36 terms in the series.
Sum of the Series (b)
To compute the sum, S, of an arithmetic series, we use the formula
S = n/2 (a1 + an). Plugging in the values we have, S = 36/2 (4 + 249) = 18 (253) = 4554.
The sum of the series is 4554.