Find the sum of the finite arithmetic sequence 6, 14, 22, 30, …, 62.
The first term of this sequence is 6 and the common difference is 14 – 6 = 8. We are summing 62 – 6 = 56 equally spaced terms. The sum is halfway between 6 + 8 and 62 – 8, i.e., $56 \cdot \frac12(6 + 62) = \boxed{1820}$.
To find the sum of a finite arithmetic sequence, you can use the formula for the sum of an arithmetic series:
Sn = (n/2)(a + l)
where Sn is the sum of the series, n is the number of terms, a is the first term, and l is the last term.
In this case, the first term (a) is 6 and the last term (l) is 62. We need to find the value of n, which represents the number of terms.
To find n, we can use the formula for the nth term of an arithmetic sequence:
an = a + (n-1)d
where an is the nth term, d is the common difference, and n is the term number. In this case, the common difference (d) is 8.
Setting l = an, we can solve for n:
62 = 6 + (n-1)8
Expanding:
62 = 6 + 8n - 8
Combining like terms:
54 = 8n - 8
Adding 8 to both sides:
62 = 8n
Dividing both sides by 8:
n = 7.75
We round up to the nearest whole number because n represents the number of terms in the sequence.
Thus, n = 8.
Now, we can substitute the values into the formula for the sum of an arithmetic sequence:
Sn = (n/2)(a + l)
= (8/2)(6 + 62)
= 4(68)
= 272
Therefore, the sum of the arithmetic sequence 6, 14, 22, 30, ..., 62 is 272.
To find the sum of a finite arithmetic sequence, you can use the formula for the sum of an arithmetic series. The formula is given by:
\[ S = \frac{n}{2}(a_1 + a_n) \]
where:
- S is the sum of the sequence,
- n is the number of terms in the sequence,
- \( a_1 \) is the first term of the sequence, and
- \( a_n \) is the nth term of the sequence.
In this case, we can identify the values:
- \( a_1 = 6 \) (the first term),
- \( a_n = 62 \) (the last term), and
- n is the number of terms that we need to determine.
To find n, we can use the formula for the nth term of an arithmetic sequence:
\[ a_n = a_1 + (n-1)d \]
where d is the common difference between consecutive terms.
Since the common difference is 8 (subtracting each term from the previous one: 14 - 6 = 8, 22 - 14 = 8, etc.), we can substitute the values into the formula to find n:
\[ 62 = 6 + (n-1)8 \]
Simplifying, we get:
\[ 62 = 6 + 8n - 8 \]
\[ 62 = 8n - 2 \]
\[ 64 = 8n \]
\[ n = 8 \]
Now that we have determined n = 8, we can calculate the sum using the formula for the sum of an arithmetic series:
\[ S = \frac{n}{2}(a_1 + a_n) \]
Substituting the values we found, we get:
\[ S = \frac{8}{2}(6 + 62) \]
\[ S = 4 \times 68 \]
\[ S = 272 \]
Therefore, the sum of the arithmetic sequence 6, 14, 22, 30, ..., 62 is 272.