Math Problem Solver

\[\sum_{x=1}^{15} \left(5 x + 430\right) = 7050\]

$$\begin{aligned}\sum_{x=1}^{15} \left(5 x + 430\right) &= \sum_{x=1}^{15} 5 x + \sum_{x=1}^{15} 430 \ \ \ \small{\color{grey}\text{by the identity} \sum_{n=k}^{m} \left(f{\left(n \right)} + g{\left(n \right)}\right) = \sum_{n=k}^{m} f{\left(n \right)} + \sum_{n=k}^{m} g{\left(n \right)}}\\&= \sum_{x=1}^{15} 5 x + 15 \cdot 430 \ \ \ \small{\color{grey}\text{by the identity} \sum_{n=1}^{m} C = mC}\\&= \sum_{x=1}^{15} 5 x + 6450\\&= 5 \sum_{x=1}^{15} x + 6450 \ \ \ \small{\color{grey}\text{by the identity} \sum_{n=k}^{m} C f{\left(n \right)} = C \sum_{n=k}^{m} f{\left(n \right)}}\\&= \frac{5 \cdot 15 \cdot \left(1 + 15\right)}{2} + 6450 \ \ \ \small{\color{grey}\text{by the identity} \sum_{n=1}^{m} n = \frac{m(m+1)}{2}}\\&= 600 + 6450\\&= 7050\end{aligned}$$