4 The sixth term of an arithmetic series is 32 and the sum of the first nine terms is 234. Determine the first three terms and the $$ n^{ ext {th }} $$ term.

Question

4 The sixth term of an arithmetic series is 32 and the sum of the first nine terms is 234. Determine the first three terms and the $$ n^{	ext {th }} $$ term.

UpStudy AI · Accepted Answer

To determine the first three terms and the $$ n^{\text{th}} $$ term of the arithmetic series, let's follow these steps:

### Given:
1. **Sixth term ($$ a_6 $$)**: 32
2. **Sum of the first nine terms ($$ S_9 $$)**: 234

### Step 1: Express the Sixth Term
The general formula for the $$ n^{\text{th}} $$ term of an arithmetic series is:
$$
a_n = a + (n-1)d
$$
For the sixth term:
$$
a_6 = a + 5d = 32 \quad \text{(1)}
$$

### Step 2: Express the Sum of the First Nine Terms
The sum of the first $$ n $$ terms is given by:
$$
S_n = \frac{n}{2} \left[ 2a + (n-1)d \right]
$$
For $$ S_9 $$:
$$
S_9 = \frac{9}{2} \left[ 2a + 8d \right] = 234 \quad \text{(2)}
$$
Simplify equation (2):
$$
2a + 8d = \frac{234 \times 2}{9} = 52
$$
$$
a + 4d = 26 \quad \text{(3)}
$$

### Step 3: Solve the System of Equations
Subtract equation (3) from equation (1):
$$
(a + 5d) - (a + 4d) = 32 - 26
$$
$$
d = 6
$$

Substitute $$ d = 6 $$ into equation (3):
$$
a + 4(6) = 26
$$
$$
a = 26 - 24 = 2
$$

### Step 4: Determine the First Three Terms
Using $$ a = 2 $$ and $$ d = 6 $$:
$$
\begin{align*}
a_1 &= a = 2 \
a_2 &= a + d = 2 + 6 = 8 \
a_3 &= a + 2d = 2 + 12 = 14 \
\end{align*}
$$
**First three terms:** 2, 8, 14

### Step 5: Find the $$ n^{\text{th}} $$ Term
Using the general formula:
$$
a_n = a + (n-1)d = 2 + 6(n-1) = 6n - 4
$$

### **Final Answer:**
- **First three terms:** 2, 8, 14
- **$$ n^{\text{th}} $$ term:** $$ a_n = 6n - 4 $$
The first three terms are 2, 8, and 14. The $$ n^{\text{th}} $$ term is $$ 6n - 4 $$.

4 The sixth term of an arithmetic series is 32 and the sum of the first nine terms is 234. Determine the first three terms and the term.

Upstudy AI Solution

Answer

Solution

Mind Expander

Related Questions

Latest Algebra Questions