The sum to $$ n $$ terms of an arithmetic series is $$ S_{n}=\frac{\pi}{2}(7 n+15 $$ 3.1 How many terms must be added to give a sum of 425 ? 3.2 Determine the sixth term of the series.

Question

UpStudy AI · Accepted Answer

To solve the problem, we start with the given sum of the first $$ n $$ terms of an arithmetic series:

$$
S_n = \frac{\pi}{2}(7n + 15)
$$

### Step 1: Find $$ n $$ for $$ S_n = 425 $$

We need to find $$ n $$ such that:

$$
\frac{\pi}{2}(7n + 15) = 425
$$

To isolate $$ n $$, we first multiply both sides by $$ \frac{2}{\pi} $$:

$$
7n + 15 = \frac{850}{\pi}
$$

Next, we subtract 15 from both sides:

$$
7n = \frac{850}{\pi} - 15
$$

Now, we divide by 7 to solve for $$ n $$:

$$
n = \frac{1}{7} \left( \frac{850}{\pi} - 15 \right)
$$

### Step 2: Calculate $$ n $$

Now, we will calculate the value of $$ n $$ using the approximate value of $$ \pi \approx 3.14 $$:

$$
n = \frac{1}{7} \left( \frac{850}{3.14} - 15 \right)
$$

Calculating $$ \frac{850}{3.14} $$:

$$
\frac{850}{3.14} \approx 270.06
$$

Now substituting back:

$$
n = \frac{1}{7} (270.06 - 15) = \frac{1}{7} (255.06) \approx 36.43
$$

Since $$ n $$ must be a whole number, we round $$ n $$ to the nearest whole number, which is 36.

### Step 3: Find the sixth term of the series

The $$ n $$-th term of an arithmetic series 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.

First, we calculate $$ S_6 $$ and $$ S_5 $$:

$$
S_6 = \frac{\pi}{2}(7 \cdot 6 + 15) = \frac{\pi}{2}(42 + 15) = \frac{\pi}{2}(57) = \frac{57\pi}{2}
$$

$$
S_5 = \frac{\pi}{2}(7 \cdot 5 + 15) = \frac{\pi}{2}(35 + 15) = \frac{\pi}{2}(50) = 25\pi
$$

Now, we find the sixth term:

$$
a_6 = S_6 - S_5 = \frac{57\pi}{2} - 25\pi
$$

To combine these, we convert $$ 25\pi $$ to have a common denominator:

$$
25\pi = \frac{50\pi}{2}
$$

Now substituting back:

$$
a_6 = \frac{57\pi}{2} - \frac{50\pi}{2} = \frac{7\pi}{2}
$$

### Final Answers

3.1 The number of terms that must be added to give a sum of 425 is approximately $$ n = 36 $$.

3.2 The sixth term of the series is $$ a_6 = \frac{7\pi}{2} $$.
3.1 Approximately 36 terms are needed to sum to 425. 3.2 The sixth term of the series is $$ \frac{7\pi}{2} $$.

The sum to \( n \) terms of an arithmetic series is \( S_{n}=\frac{\pi}{2}(7 n+15 \) 3.1 How many terms must be added to give a sum of 425 ? 3.2 Determine the sixth term of the series.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Extra Insights

preguntas relacionadas

Latest Arithmetic Questions