the arithmetic series 1 + 4 + 7 + ....tn has a sum of 1001. how many terms does the series have?

To find the number of terms in an arithmetic series, we need to use the formula for the sum of an arithmetic series, which is given by:

Sn = (n/2) * (2a + (n-1)d)

Where:
Sn is the sum of the series,
n is the number of terms,
a is the first term, and
d is the common difference.

We are given that the sum of the series is 1001. We also know that the first term (a) is 1 and the common difference (d) is 3 (since each term increases by 3).

Plugging in the given information into the formula, we have:

1001 = (n/2) * (2*1 + (n-1)*3)

We can simplify this equation:

1001 = (n/2) * (2 + 3n - 3)

1001 = (n/2) * (3n - 1)

Now, let’s solve this equation to find n.

Multiplying both sides of the equation by 2 to eliminate the fraction:

2002 = n(3n - 1)

Rearranging the equation:

3n^2 - n - 2002 = 0

To solve this quadratic equation, we can either factor it, use the quadratic formula, or solve it graphically. By factoring, we get:

(3n + 59)(n - 34) = 0

Setting each factor equal to zero and solving for n, we have:

3n + 59 = 0 or n - 34 = 0

n = -59/3 or n = 34

Since the number of terms cannot be negative, we take n = 34 as the solution.

Therefore, the arithmetic series has 34 terms.

To find out how many terms there are in the arithmetic series, we need to determine the common difference and use the formula for the nth term of an arithmetic sequence.

In this case, we can observe that the common difference (d) is 4 - 1 = 3. So, each term increases by 3.

The formula for the nth term of an arithmetic sequence is: tn = a + (n-1) * d, where a is the first term, n is the number of terms, and d is the common difference.

Since the first term (a) is 1, we have the equation: tn = 1 + (n-1) * 3.

Given that the sum of the series is 1001, we can use the formula for the sum of an arithmetic series: Sn = (n/2) * (a + tn). In this case, Sn = 1001.

We can substitute the variables into the equation:

1001 = (n/2) * (1 + (1 + (n-1) * 3))

Simplifying the equation, we get:

1001 = (n/2) * (2 + 3n - 3)

1001 = (n/2) * (3n - 1)

Now, we can solve this quadratic equation. Rearranging it, we get:

(n/2) * (3n - 1) = 1001

Multiplying both sides by 2 to eliminate the fraction:

n * (3n - 1) = 2002

Expanding the equation, we get:

3n² - n - 2002 = 0

Using the quadratic formula, n = (-b ± √(b² - 4ac)) / (2a), where a = 3, b = -1, and c = -2002, we can find the value(s) of n.

n = (1 ± √((1²) - 4 * 3 * -2002)) / (2 * 3)

After calculating, we find that n ≈ 17.01 or n ≈ -39.01.

Since the number of terms cannot be negative, the series has approximately 17 terms.

apply arithmetic progression formula !If you don't know then search arithmetic progression on GOOGLE!