Question

The first term of an arithmetic sequence is equal to 200 and the common difference is equal to -10. Find the value of the ${20^{th}}$ term.

Answer 1

We will use the formula to determine the ${n^{th}}$ term of an arithmetic progression (A. P.) given by: ${a_n} = a + \left( {n - 1} \right)d$ , where a is the first term of the A. P. and d is the common difference. Upon substituting the values of a, n and d, we will get the value of ${20^{th}}$ term.

Complete step-by-step answer:
We are given the first term, of an arithmetic sequence as 200.
Its common difference, d is -10.
We need to calculate the value of the ${20^{th}}$ term.
We know that the formula to calculate the ${n^{th}}$ term of an A. P. is given as: ${a_n} = a + \left( {n - 1} \right)d$, where a is the first term of the sequence and d is the common difference of the A. P.
Here, n = 20, a = 200 and d = –10. Substituting these values in the equation ${a_n} = a + \left( {n - 1} \right)d$, we get
$\ \Rightarrow {a_n} = a + \left( {n - 1} \right)d \\\ \Rightarrow {a_{20}} = 200 + \left( {20 - 1} \right)\left( { - 10} \right) \\\ \Rightarrow {a_{20}} = 200 - 10\left( {19} \right) \\\ \Rightarrow {a_{20}} = 200 - 190 \\\ \Rightarrow {a_{20}} = 10 \\\ \$
Therefore, the value of the ${20^{th}}$ term of the arithmetic sequence whose first term is 200 and common difference is –10 is calculated to be 10.

Note: In this question, we have only used the formula of ${n^{th}}$ term by simply putting the values given in the question. It is a straightforward question based on direct formula. You may go wrong only in the determination of the value of the ${20^{th}}$ term (in terms of the calculation part).
In mathematics, an arithmetic progression is a sequence of numbers such that the difference between two consecutive terms is always a constant (termed as common difference). For example, 2, 4, 6, 8, 10, 12, … is an arithmetic progression with a common difference of 2.