Find the sum of the arithmetic series given a_1=2, a_n=35 , and n=12. A. 226 B. 225 C. 222 D. 444
The correct answer and explanation is:
To find the sum of the arithmetic series, you can use the formula for the sum of an arithmetic series: Sn=n2⋅(a1+an)S_n = \frac{n}{2} \cdot (a_1 + a_n)
Where:
- SnS_n is the sum of the first nn terms.
- a1a_1 is the first term of the series.
- ana_n is the nth term of the series.
- nn is the number of terms in the series.
Given:
- a1=2a_1 = 2
- an=35a_n = 35
- n=12n = 12
Now, substitute these values into the formula: S12=122⋅(2+35)S_{12} = \frac{12}{2} \cdot (2 + 35) S12=6⋅37S_{12} = 6 \cdot 37 S12=222S_{12} = 222
Thus, the sum of the arithmetic series is 222.
Explanation:
The sum of an arithmetic series is based on the concept that the series progresses in a regular, linear pattern. The first term is 2, and the 12th term is 35. By averaging the first and last terms of the series (which is a1+ana_1 + a_n), and then multiplying by the number of terms divided by 2, we can efficiently compute the total sum. This method simplifies the process of adding each individual term in the series, making it faster and more efficient for larger sequences. In this case, the sum of the 12 terms is 222, so the correct answer is C.