Chapter 7.2, Problem 58E

Solution

To calculate: The value of $n$ if summation is ${\displaystyle \sum _{i=1}^n\left(5-5i\right)=-50}$ .

The value of $n=5$ .

Given information:

The given terms are ${\displaystyle \sum _{i=1}^n\left(5-5i\right)=-50}$ .

Formula used:

The sum nth term of an arithmetic sequence with first term $a_1$ and $a_n$ is given by:

  $s_n=n\left(\frac{a_1+\left(a_1+\left(n-1\right)d\right)}{2}\right)$

Calculation:

Consider the given sequence.

It is given that ${\displaystyle \sum _{i=1}^n\left(5-5i\right)=-50}$ .

In above seen that $a_n=5-5n$ .

  $\begin{array}{c}{a}_n=5-5n\\ a_n=5-5n+5-5\\ a_n=-5\left(n-1\right)\end{array}$

The nth term of an arithmetic sequence with first term $a_1$ and common difference $d$ is given by: $a_n=a_1+\left(n-1\right)d$ .

  $a_1=0\text{and}d=-5$.

  $\begin{array}{c}{s}_n=n\left(\frac{a_1+\left(a_1+\left(n-1\right)d\right)}{2}\right)\\ -50=n\left(\frac{0+\left(0+\left(n-1\right)\left(-5\right)\right)}{2}\right)\\ -100=+5n-5n^2\end{array}$

Now solve this quadratic equation using the quadratic formula.

  $\begin{array}{c}n=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ n=\frac{-5\pm \sqrt{5^2-4\cdot \left(-5\right)\cdot 100}}{2\cdot \left(-5\right)}\\ n=\frac{-5\pm \sqrt{2025}}{-10}\\ n=\frac{-5\pm 45}{-10}\\ n=\frac{-5+45}{-10}\text{or}n=\frac{-5-45}{-10}\\ n=\frac{40}{-10}\text{or}n=\frac{-50}{-10}\\ n=-4\text{or}n=5\end{array}$

Since the solution is positive and integer thus the only result is $5$ .