Math Problem Solver

The sum of the sequence is \(\displaystyle 3825\)

This is an arithmetic sequence.

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

Explicit Formula: $$a_n=n + 49$$

Recursive Formula: $$a_n=a_{n-1} + 1, \text{where } a_1=50$$

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:

$$\begin{aligned} S_n&=\sum_{i=1}^{n} a_{i} \\ &=\sum_{i=1}^{n} \left(i + 49\right) \\ &= \sum_{i=1}^{n} 49 + \sum_{i=1}^{n} i \ \ \ \small{\color{grey}\text{by the identity} \sum_{n=k}^{m} \left(f{\left(n \right)} + g{\left(n \right)}\right) = \sum_{n=k}^{m} f{\left(n \right)} + \sum_{n=k}^{m} g{\left(n \right)}}\\&= \sum_{i=1}^{n} i + 49 n \ \ \ \small{\color{grey}\text{by the identity} \sum_{n=1}^{m} C = mC}\\&= 49 n + \frac{n \left(1 + n\right)}{2} \ \ \ \small{\color{grey}\text{by the identity} \sum_{n=1}^{m} n = \frac{m(m+1)}{2}}\\&=\frac{n \left(n + 99\right)}{2} \end{aligned}$$where $n$ is the is the term number.

Use the explicit formula to find the index of the final term in the sequence.$$\begin{aligned}100 &= n + 49 \\ n &= 51\end{aligned}$$Then use the summation formula to solve.$$\begin{aligned}S_n &= \frac{n \left(n + 99\right)}{2} \\S_{51} &= \frac{51 \cdot \left(99 + 51\right)}{2}\\S_{51} &= 3825\end{aligned}$$