If the sum of the first n terms of the series 4 + 7 + 10 + ... is 209, find n.
n/2 (2*4 + (n-1)*3) = 209
1/2 (3n^2+5n) = 209
3n^2+5n-418 = 0
(3n+38)(n-11) = 0
n=11
Very helpful and simplify. Thanks
209
Why did the number go to therapy? Because it had too many issues! But to solve this problem, let's first find the common difference between consecutive terms. We can see that each term increases by 3. So the nth term can be represented by the equation Tn = 4 + 3(n-1). Now, we need to find the number of terms (n) where the sum of the first n terms is 209. Let's plug in the sum equation:
209 = (n/2)(2(4) + (n-1)(3))
Now let's simplify this expression:
209 = (n/2)(8 + 3n - 3)
209 = (n/2)(5 + 3n)
Okay, let's distribute the n/2 to the terms inside the parenthesis:
209 = (5n/2) + (3n²/2)
Now we have a quadratic equation. Let's multiply everything by 2 to make the equation cleaner:
418 = 5n + 3n²
Rearranging the equation:
3n² + 5n - 418 = 0
After solving the quadratic equation, we find that n is approximately equal to 12. So, the sum of the first 12 terms in the series 4 + 7 + 10 + ... is 209.
The answer is 11.
To find the value of n, we need to determine the number of terms in the series.
The given series is an arithmetic progression with a common difference of 3. The general formula for the nth term of an arithmetic progression is given by:
an = a1 + (n - 1) * d
where:
an represents the nth term,
a1 represents the first term, and
d represents the common difference.
In this case, the first term a1 is 4 and the common difference d is 3.
Now we can use the formula for the sum of an arithmetic series to find n:
Sn = (n/2) * (a1 + an)
Given that the sum of the first n terms (Sn) is 209, we can substitute the known values into the formula:
209 = (n/2) * (4 + a1 + (n - 1) * d)
Simplifying further:
209 = (n/2) * (4 + 4 + 3n - 3)
Now, we can distribute and combine like terms:
209 = (n/2) * (7 + 3n)
Multiply both sides of the equation by 2 to eliminate the fraction:
418 = n*(7 + 3n)
Rearranging the equation to form a quadratic equation:
3n^2 + 7n - 418 = 0
Now we can solve this quadratic equation to find the possible values of n.