The first two terms of an arithmetic series are
-
2
and 3. How many terms are needed for the sum to equal 306 ?
Let's call the common difference in the arithmetic series d. In this case, d = 3 - (-2) = 5. The formula for the sum of an arithmetic series is given by:
Sum = n*(a1 + an) / 2,
where a1 is the first term, an is the nth term, and n is the number of terms.
We know that the first term a1 = -2, and we have to find the number of terms n such that Sum = 306.
We also know that an = a1 + (n-1) * d. Substituting a1 and d in this formula, we get:
an = -2 + (n-1) * 5.
Now we can substitute a1 and an in the formula for the sum:
306 = n*(-2 + (-2 + (n - 1) * 5)) / 2.
Simplifying and solving for n, we get:
306 = n*(n*5 - 4) / 2.
612 = n*(5n - 4).
At this point, you may try out different values of n to see which one satisfies the above equation. If you try n = 6, you'll get 612 = 6 * 26, which is true. Therefore, n = 6 is the answer.
So, 6 terms are needed for the sum to equal 306.
To find the number of terms needed for the sum to equal 306 in an arithmetic series, we can use the formula for the sum of an arithmetic series:
S = (n/2)(a + l)
Where S is the sum of the series, n is the number of terms, a is the first term, and l is the last term.
Given that the first term a = -2, we need to find the last term, l. To do this, we need to determine the common difference, d, of the series.
The common difference, d, can be found by subtracting the first term from the second term:
d = 3 - (-2)
= 5
Now we can find the last term, l, using the formula for the nth term of an arithmetic series:
l = a + (n - 1)d
Substituting in the values we have:
l = -2 + (n - 1)5
l = -2 + 5n - 5
l = 5n - 7
Now we can substitute the values of a and l into the sum formula to solve for n:
306 = (n/2)(-2 + (5n - 7))
To simplify:
306 = (n/2)(5n - 9)
Next, we can multiply both sides by 2 to eliminate the fraction:
612 = n(5n - 9)
Now we have a quadratic equation:
5n^2 - 9n - 612 = 0
Solve this equation using factoring, completing the square, or the quadratic formula to find the value of n.
To find the number of terms needed for the sum to equal 306 in an arithmetic series, we will follow these steps:
Step 1: Find the common difference (d):
The common difference is the difference between any two consecutive terms in an arithmetic series.
In this case, the first term is -2 and the second term is 3.
So, the common difference (d) = 3 - (-2) = 5.
Step 2: Find the formula for the sum (Sn) of the arithmetic series:
The formula for the sum of an arithmetic series is given by:
Sn = (n/2) * (2a + (n-1)d),
where n is the number of terms, a is the first term, and d is the common difference.
Step 3: Substitute the known values into the formula and solve for n:
We have Sn = 306, a = -2, and d = 5.
306 = (n/2) * (2*(-2) + (n-1)*5).
306 = (n/2) * (-4 + 5n - 5).
306 = (n/2) * (5n - 9).
Step 4: Solve the equation for n:
Multiply both sides of the equation by 2 to eliminate the fraction:
612 = n(5n - 9).
Distribute n on the right side of the equation:
612 = 5n^2 - 9n.
Rearrange the equation to standard form:
5n^2 - 9n - 612 = 0.
Step 5: Solve the quadratic equation:
Factor or use the quadratic formula to solve for n:
In this case, we will use the quadratic formula:
n = (-b ± √(b^2 - 4ac)) / 2a,
where a = 5, b = -9, and c = -612.
n = (-(-9) ± √((-9)^2 - 4*5*(-612))) / (2*5).
n = (9 ± √(81 + 12240)) / 10.
n = (9 ± √12321) / 10.
Simplifying the square root:
n = (9 ± 111) / 10.
Therefore, we have two possible solutions for n:
1. n = (9 + 111) / 10 = 120 / 10 = 12.
2. n = (9 - 111) / 10 = -102 / 10 = -10.2.
Since the number of terms cannot be negative, the only valid solution is n = 12.
So, 12 terms are needed for the sum to equal 306 in the given arithmetic series.