Question

A1=429 d=-22 S(n)=3069 n-?

Answer 1

A1=429
d=-22
Sn=-3096.

Sn=(2a1-d(n-1))n/2.
-3096=(2*429+22(n-1))n/2.  n = 239

Answer 2

To find the value of 'n' in the arithmetic sequence, we can use the formula for the nth term of an arithmetic sequence:

A_n = A_1 + (n - 1) * d

Where: A_n is the nth term, A_1 is the first term, n is the number of terms, and d is the common difference.

In this case, we know: A_1 = 429, d = -22, and S(n) = 3069.

The sum of the first 'n' terms of an arithmetic sequence can be calculated using the formula:

S(n) = (n/2) * (2*A_1 + (n - 1) * d)

Substituting the given values:

3069 = (n/2) * (2 * 429 + (n - 1) * -22)

Now, we can solve this equation to find the value of 'n'. Let's simplify it:

3069 = (n/2) * (858 - 22n + 22) 3069 = (n/2) * (880 - 22n)

Multiplying both sides by 2 to remove the fraction:

6138 = n * (880 - 22n)

Expanding:

6138 = 880n - 22n^2

Rearranging and setting the equation to zero:

22n^2 - 880n + 6138 = 0

Now, we can solve this quadratic equation using the quadratic formula:

n = (-b ± √(b^2 - 4ac)) / (2a)

For this equation, a = 22, b = -880, and c = 6138. Let's substitute these values:

n = (-(-880) ± √((-880)^2 - 4 * 22 * 6138)) / (2 * 22) n = (880 ± √(774400 - 540312)) / 44 n = (880 ± √234088) / 44 n = (880 ± 483.791) / 44

Now, let's calculate the two possible solutions for 'n':

n_1 = (880 + 483.791) / 44 n_1 = 1363.791 / 44 n_1 ≈ 30.99

n_2 = (880 - 483.791) / 44 n_2 = 396.209 / 44 n_2 ≈ 9.00

Since 'n' represents the number of terms in the sequence, it cannot be a decimal or fraction. Therefore, the value of 'n' is approximately 9.