To find the common difference of an arithmetic series where the first term is 2, the last term is 29, and the sum of the series is 155, we can use the formula for the sum of an arithmetic series and the formula for the nth term of an arithmetic series.
The formula for the sum [tex]\( S \)[/tex] of an arithmetic series is:
[tex]\[ S = \frac{n}{2} (a_1 + a_n) \][/tex]
where:
- [tex]\( S \)[/tex] is the sum of the series,
- [tex]\( n \)[/tex] is the number of terms,
- [tex]\( a_1 \)[/tex] is the first term,
- [tex]\( a_n \)[/tex] is the last term.
The formula for the nth term [tex]\( a_n \)[/tex] of an arithmetic series is:
[tex]\[ a_n = a_1 + (n - 1)d \][/tex]
where:
- [tex]\( a_1 \)[/tex] is the first term,
- [tex]\( n \)[/tex] is the nth term,
- [tex]\( d \)[/tex] is the common difference.
We have [tex]\( a_1 = 2 \)[/tex], [tex]\( a_n = 29 \)[/tex], and [tex]\( S = 155 \)[/tex]. First, we can find the number of terms [tex]\( n \)[/tex] using the sum formula:
[tex]\[ 155 = \frac{n}{2} (2 + 29) \][/tex]
[tex]\[ 155 = \frac{n}{2} (31) \][/tex]
[tex]\[ 155 \times 2 = n \times 31 \][/tex]
[tex]\[ 310 = 31n \][/tex]
[tex]\[ n = \frac{310}{31} \][/tex]
[tex]\[ n = 10 \][/tex]
So, there are 10 terms in the series.
Now that we have [tex]\( n \)[/tex], we can use the nth term formula to find the common difference [tex]\( d \)[/tex]:
[tex]\[ 29 = 2 + (10 - 1)d \][/tex]
[tex]\[ 29 = 2 + 9d \][/tex]
[tex]\[ 29 - 2 = 9d \][/tex]
[tex]\[ 27 = 9d \][/tex]
[tex]\[ d = \frac{27}{9} \][/tex]
[tex]\[ d = 3 \][/tex]
Therefore, the common difference [tex]\( d \)[/tex] of the arithmetic series is 3.
