The sixth term of an arithmetic progression is 265 and the sum of the first 5 terms is 1445. Find the minimum value of n so that the sum of the first n terms is negative.

a+5d = 265

5/2 (2a+4d) = 1445

(a,d) = (305,-8)

So, we want

n/2 (2a+(n-1)d) < 0
n/2 (2*305 + (n-1)(-8)) < 0
n > 77.25

so, you need 78 terms before the sum is negative

makes sense, since the 39th term is negative, so you need another 39 terms to offset the first 39.

I canโ€™t really understand why need to be 78๐Ÿ˜“

To find the minimum value of n such that the sum of the first n terms is negative, we need to determine the sum of the first n terms and check when it becomes negative.

Let's start by finding the common difference (d) of the arithmetic progression. We know that the sixth term (aโ‚†) is 265, and the sum of the first 5 terms (Sโ‚…) is 1445.

The formula for finding the nth term of an arithmetic progression is:
aโ‚™ = aโ‚ + (n-1)d --- (1)

The formula for the sum of the first n terms of an arithmetic progression is:
Sโ‚™ = (n/2)(2aโ‚ + (n-1)d) --- (2)

From equation (1), we can substitute aโ‚ with aโ‚† - 5d:
aโ‚ = aโ‚† - 5d

Substituting this value into equation (2), we have:
Sโ‚… = (5/2)(2(aโ‚† - 5d) + (5-1)d)
1445 = 2aโ‚† - 10d + 2d
1445 = 2aโ‚† - 8d --- (3)

We also know that the sixth term (aโ‚†) is 265:
aโ‚† = 265

Substituting this value into equation (3), we get:
1445 = 2(265) - 8d
1445 = 530 - 8d
8d = 530 - 1445
8d = -915
d = -915 / 8
d โ‰ˆ -114.375

Now that we have the common difference, we can find the formula for the sum of the first n terms of the arithmetic progression using equation (2):
Sโ‚™ = (n/2)(2aโ‚ + (n-1)d)

Substituting aโ‚ with aโ‚† - 5d and aโ‚† with 265:
Sโ‚™ = (n/2)(2(265 - 5d) + (n - 1)d)
Sโ‚™ = (n/2)(530 - 10d + nd - d)
Sโ‚™ = (n/2)(530 + (n - 11)d) --- (4)

To find the minimum value of n for which the sum of the first n terms is negative, we need to find the first value of n that makes the sum of the first n terms negative.

Substituting the given values into equation (4), we get:
Sโ‚™ = (n/2)(530 + (n - 11)(-114.375))
Sโ‚™ = (n/2)(530 - 114.375n + 1258.125)
Sโ‚™ = (n/2)(1258.125 - 114.375n)
Sโ‚™ = (n(1258.125 - 114.375n))/2

We want to find the minimum value of n such that Sโ‚™ becomes negative. So, we need to find the value of n that makes (1258.125 - 114.375n) negative.

1258.125 - 114.375n < 0
114.375n > 1258.125
n > 1258.125 / 114.375
n > 11

Since n must be an integer, the minimum value of n such that the sum of the first n terms is negative is n = 12.