A partial sum of an arithmetic sequence is given. Find the sum.
1 + 7 + 13 + ...... + 607
S_101 = 101/2 (1 + 607) = ___
To find the sum of an arithmetic sequence, you need to know the first term, the common difference, and the number of terms. However, in this case, we are given the partial sum instead. Therefore, we need to work backward to find the sum.
The partial sum formula for an arithmetic sequence is given by:
Sn = (n/2)(a1 + an),
where Sn is the sum of the first n terms, a1 is the first term, and an is the nth term.
In this case, Sn = 607 (the given partial sum).
Let's substitute the values into the formula and solve for n:
607 = (n/2)(1 + an).
We know that the common difference in an arithmetic sequence is constant, so let's find the common difference:
d = a2 - a1,
where d is the common difference.
Given a1 = 1, we can find a2:
a2 = a1 + d = 1 + (7 - 1) = 1 + 6 = 7.
Therefore, d = a2 - a1 = 7 - 1 = 6.
Now, let's substitute the values into the formula and solve for n:
607 = (n/2)(1 + 7).
607 = 4n.
Divide both sides of the equation by 4:
607/4 = n.
n = 151.75.
Since the number of terms (n) must be a positive integer, we round up to the nearest whole number:
n = 152.
Now, we can find the sum of the arithmetic sequence:
Sn = (n/2)(a1 + an).
Sn = (152/2)(1 + an).
Using the formula for the nth term:
an = a1 + (n-1)d,
where d is the common difference:
an = 1 + (152-1)(6).
an = 1 + 151(6).
an = 1 + 906.
an = 907.
Substituting the values into the formula for the sum:
Sn = (152/2)(1 + 907).
Sn = 76(908).
Sn = 69,248.
Therefore, the sum of the arithmetic sequence is 69,248.
To find the sum of an arithmetic sequence, you can use the formula for the sum of a finite arithmetic series:
Sn = (n/2)(a + L)
Where:
- Sn is the sum of the arithmetic sequence,
- n is the number of terms in the sequence,
- a is the first term of the sequence, and
- L is the last term of the sequence.
In your case, the given sequence has a common difference of 6. So, the first term, a, is 1, and the last term, L, is 607.
To find the number of terms, n, we need to determine the position of the term 607 in the sequence. The formula to find the nth term of an arithmetic sequence is:
an = a + (n - 1)d
Where:
- an is the nth term of the sequence,
- a is the first term of the sequence,
- n is the position of the term in the sequence,
- d is the common difference.
To find n, let's substitute the given values into the formula:
607 = 1 + (n - 1)6
Now, solve for n:
606 = (n - 1)6
n - 1 = 606/6
n - 1 = 101
n = 102
Now that we have the value of n, we can calculate the sum, Sn:
Sn = (n/2)(a + L)
= (102/2)(1 + 607)
= 51(608)
= 31,008
Therefore, the sum of the arithmetic sequence 1 + 7 + 13 + ...... + 607 is 31,008.