Find n if S(n) = 23,001 for the arithmetic series 4+7+10...

To find the value of n for the arithmetic series, we first need to find the formula for the nth term (S(n)) of the series.

In an arithmetic series, each term is obtained by adding a constant number, called the common difference (d), to the previous term.

In this case, the first term, a, is 4, and the common difference, d, is 7 - 4 = 3.

The formula for the nth term of an arithmetic series is given by:

An = a + (n - 1) * d

where An represents the nth term.

Now, we have S(n) = 23,001 and we need to find the value of n.

The sum of an arithmetic series is given by the formula:

S(n) = (n/2) * (a + L)

where L represents the last term of the series.

In this case, we need to find the value of n when S(n) is 23,001.

So, we can set up the equation:

23,001 = (n/2) * (a + L)

Let's find the values of a and L:

The first term, a, is given as 4.

To find the last term, L, we need to know the sequence of numbers in the series. The given series is 4, 7, 10, ...

We can find the last term by finding the nth term of the series and determining when it exceeds 23,001.

Let's set up an equation for the nth term and solve for n:

An = a + (n - 1) * d

23,001 = 4 + (n - 1) * 3

Now, we need to solve this equation for n.

23,001 = 4 + 3n - 3

Collecting like terms:

23,001 = 3n + 1

Subtracting 1 from both sides:

23,000 = 3n

Dividing both sides by 3:

n = 23,000 / 3

Calculating this:

n = 7,666.67

Since n represents the number of terms in the series, it cannot be a decimal or fraction. Thus, the value of n would round down to the nearest whole number, which is 7,666.

Therefore, the value of n for the arithmetic series 4 + 7 + 10 + ... such that S(n) = 23,001 is 7,666.