Math Problem Solver

The sum of the sequence is $\displaystyle 8043$

This is an arithmetic sequence.

The nth term in this sequence is given by the formula:

Explicit Formula: $$a_n=10 n + 273$$

Recursive Formula: $$a_n=a_{n-1} + 10, \text{where } a_1=283$$

Summation Formula:

Option 1:

$$S_n=\frac{n}{2} (2 a_1 + (n - 1) d)$$ where $a_1$ is the 1^st term, $d$ is the common difference, and $n$ is the is the term number.

Option 2:

$$S_n=n \left(\frac{a_1 + a_n}{2}\right)$$ where $a_1$ is the 1^st term, $a_n$ is the n^th term, and $n$ is the is the term number.

Option 3:

$$\begin{aligned} S_n&=\sum_{i=1}^{n} a_{i} \\ &=\sum_{i=1}^{n} \left(10 i + 273\right) \\ &= \sum_{i=1}^{n} 273 + \sum_{i=1}^{n} 10 i \ \ \ \small{\color{grey}\text{by the identity} \sum_{n=k}^{m} \left(f{\left(n \right)} + g{\left(n \right)}\right) = \sum_{n=k}^{m} f{\left(n \right)} + \sum_{n=k}^{m} g{\left(n \right)}}\\&= \sum_{i=1}^{n} 10 i + 273 n \ \ \ \small{\color{grey}\text{by the identity} \sum_{n=1}^{m} C = mC}\\&= 10 \sum_{i=1}^{n} i + 273 n \ \ \ \small{\color{grey}\text{by the identity} \sum_{n=k}^{m} C f{\left(n \right)} = C \sum_{n=k}^{m} f{\left(n \right)}}\\&= 273 n + \frac{10 n \left(1 + n\right)}{2} \ \ \ \small{\color{grey}\text{by the identity} \sum_{n=1}^{m} n = \frac{m(m+1)}{2}}\\&=n \left(5 n + 278\right) \end{aligned}$$where $n$ is the is the term number.

Use the explicit formula to find the index of the final term in the sequence.$$\begin{aligned}483 &= 10 n + 273 \\ n &= 21\end{aligned}$$Then use the summation formula to solve.$$\begin{aligned}S_n &= n \left(5 n + 278\right) \\S_{21} &= 21 \cdot \left(278 + 5 \cdot 21\right)\\S_{21} &= 8043\end{aligned}$$