(c) The sum of the first $$ n $$ terms of an arithmetic series is $$ \mathrm{S}_{n}=\frac{n}{2}(3-5 n) $$. (1) Determine the general term of the sequence. (2) Hence, determine the value of the $$ 40^{ ext {th }} $$ term.

Question

(c) The sum of the first $$ n $$ terms of an arithmetic series is $$ \mathrm{S}_{n}=\frac{n}{2}(3-5 n) $$. (1) Determine the general term of the sequence. (2) Hence, determine the value of the $$ 40^{	ext {th }} $$ term.

UpStudy AI · Accepted Answer

Let's solve the problem step by step.

### Given:
The sum of the first $$ n $$ terms of an arithmetic series is:
$$
\mathrm{S}_n = \frac{n}{2}(3 - 5n)
$$

### (1) Determine the General Term of the Sequence ($$ a_n $$):

For an arithmetic sequence, the $$ n $$-th term $$ a_n $$ can be found using the formula:
$$
a_n = S_n - S_{n-1}
$$
where $$ S_n $$ is the sum of the first $$ n $$ terms and $$ S_{n-1} $$ is the sum of the first $$ n-1 $$ terms.

**Step-by-Step Calculation:**

1. **Express $$ S_n $$ and $$ S_{n-1} $$:**
   $$
   S_n = \frac{n}{2}(3 - 5n)
   $$
   $$
   S_{n-1} = \frac{n-1}{2}(3 - 5(n-1)) = \frac{n-1}{2}(3 - 5n + 5) = \frac{n-1}{2}(8 - 5n)
   $$

2. **Compute $$ a_n $$:**
   $$
   a_n = S_n - S_{n-1} = \frac{n}{2}(3 - 5n) - \frac{n-1}{2}(8 - 5n)
   $$
   $$
   = \frac{1}{2} \left[ n(3 - 5n) - (n-1)(8 - 5n) \right]
   $$
   $$
   = \frac{1}{2} \left[ 3n - 5n^2 - 8n + 8 + 5n^2 - 5n \right]
   $$
   $$
   = \frac{1}{2} (-10n + 8) = -5n + 4
   $$

**Conclusion:**
$$
a_n = -5n + 4
$$

### (2) Determine the Value of the $$ 40^{\text{th}} $$ Term ($$ a_{40} $$):

Using the general term formula:
$$
a_{40} = -5(40) + 4 = -200 + 4 = -196
$$

**Final Answer:**
1. The general term is $$ a_n = -5\,n + 4 $$.
2. The $$40^{\text{th}}$$ term is –196.
1. The general term of the sequence is $$ a_n = -5n + 4 $$. 2. The 40th term is -196.

© The sum of the first terms of an arithmetic series is .
(1) Determine the general term of the sequence.
(2) Hence, determine the value of the term.

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Algebra Questions

© The sum of the first terms of an arithmetic series is . (1) Determine the general term of the sequence. (2) Hence, determine the value of the term.