Find the sum of the following (starts with 3, increment by 7, ends with 7N+3)

3+10+17+...+689+696+703

the term of 7N+3 tells me that you want the sum of n terms

sum(n) = (n/2)(2a + (n-1)d )
sum(n) = (n/2)(6 + (n-1)(7))
= (n/2)(6 + 7n - 7)
= n(7n - 1)/2

notice that ... (703 + 3) = (696 + 10) = (689 + 17) ...

so ... sum = (703 + 3) * (101 / 2)

How do you know n is 101?

703 - 3 = 700 ... 700 / 7 = 100

100 terms beyond the 1st term ... a total of 101

To find the sum of the given sequence, we can use the formula for the sum of an arithmetic series.

The first term (a) of the sequence is 3, and the common difference (d) is 7. The last term (l) can be expressed as 7N + 3.

We need to find the value of N when the last term is given. From the given sequence, the last term is 703. By substituting this value into the equation, we can solve for N:

7N + 3 = 703
7N = 703 - 3
7N = 700
N = 100

Now that we have the value of N, we can calculate the sum of the sequence using the formula:

S = (n/2)(2a + (n-1)d)

where S is the sum, n is the number of terms, a is the first term, and d is the common difference.

To find the number of terms (n), we can use the formula:

n = (l - a) / d + 1

Plugging in the known values:

n = (703 - 3) / 7 + 1
n = 100 + 1
n = 101

Now, we can calculate the sum:

S = (101/2)(2(3) + (101-1)(7))
S = (101/2)(6 + 100(7))
S = (101/2)(6 + 700)
S = (101/2)(706)
S = 101 * 353
S = 35,653

Therefore, the sum of the given sequence is 35,653.