Question

How do you find d for the arithmetic series with $S_{17}=-170$ and $a_1=2$?

Answer 1

An arithmetic series is a sequence in which all the numbers in the series are in a definite pattern. In an arithmetic sequence if we consider two consecutive numbers then the difference between them is a constant which can also be called a common difference. The generalized form of terms of an arithmetic series is ${{a}_{n}}={{a}_{1}}+\left( n-1 \right)d$, this gives the nth term where ${{a}_{1}}$ is the first term and d is the common difference. The generalized form of sum of n terms of an arithmetic sequence is ${{s}_{n}}=\dfrac{n}{2}\left( 2{{a}_{1}}+\left( n-1 \right)d \right)$.

Complete step by step answer:
As per the given question, we need to find the value of d by using the given values.
Given $S_{17}=-170$, $a_1=2$ on substituting these values in the generalized form of sum of n terms of arithmetic sequence. Since we have given value ${{s}_{17}}$ then the number of terms will be 17.
We get the value as $$$$

& {{s}_{17}}=\dfrac{17}{2}\left( 2\left( 2 \right)+\left( 17-1 \right)d \right) \\\ & -170=\dfrac{17}{2}\left( 2\left( 2 \right)+\left( 17-1 \right)d \right) \\\ \end{aligned}$$ $$\begin{aligned} & -170=\dfrac{17}{2}\left( 4+16d \right) \\\ & -170\times 2=17(4+16d) \\\ & -340=17(4+16d) \\\ \end{aligned}$$ We know that -340 is a multiple of 17. So we can divide -340 with 17 then we get the value of division as -20. $$-20=4+16d$$ since we know that 16 is a multiple of 4 we can take common from both the terms.$$d=-1.5$$ $$-20=4(1+4d)$$ Since -20 is also a multiple of 4. We divide -20 with 4 then we get the value of division as -5. $$-5=1+4d$$ On adding -1 on both sides. We get the value as $$-6=4d$$ On solving the above equation as $$d=-1.5$$ **Therefore, the common difference d for the arithmetic series with $$S_{17}=-170$$ and $$a_1=2$$ is $$-1.5$$.** **Note:** While solving questions from arithmetic sequence one common error would be not correctly finding the value of d, the common difference. Sometimes sequences of fractions are confusing. You might check that the d calculated is consistently true for any two successive terms of the sequence. This helps to verify the sequence.