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 20th term.

Answer 1

Here, we will find the common difference and then use the formula of $n$th term of the arithmetic progression A.P., that is, ${a_n} = a + \left( {n - 1} \right)d$, where $a$ is the first term and $d$ is the common difference. Apply this formula, and then substitute the value of $a$,$d$ and $n$ in the obtained equation to find the value of the required term.

Complete step by step answer:

We are given that the first term of an arithmetic sequence is 200 and the common difference is $- 10$.

We know that the arithmetic progression is a sequence of numbers in order in which the difference of any two consecutive numbers is a constant value.

We will now find the value of first term $a$ and the common difference$d$, we get
$a = 200$
$d = - 10$

We will use the formula of $n$th term of the arithmetic progression A.P., that is, ${a_n} = a + \left( {n - 1} \right)d$, where $a$ is the first term and $d$ is the common difference.

We know that $n = 20$.
Substituting these values of $n$, $a$ and $d$ in the above formula for the sum of the arithmetic progression, we get

$$\Rightarrow {a_{20}} = 200 + \left( {20 - 1} \right)\left( { - 10} \right) \\\ \Rightarrow {a_{20}} = 200 + 19\left( { - 10} \right) \\\ \Rightarrow {a_{20}} = 200 - 190 \\\ \Rightarrow {a_{20}} = 10 \\\$$

Thus, the 20th term of the given AP is 10.

Note: In solving these types of questions, you should be familiar with the formula of sum of the arithmetic progression and their sums. Some students use the formula to find the sum, $S = \dfrac{n}{2}\left( {a + l} \right)$, where $l$ is the last term, but we do not have the to find the value of ${a_n}$ so it will be wrong. We can also find the value of $n$th term by find the value of ${S_n} - {S_{n - 1}}$, where ${S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$, where $a$ is the first term and $d$ is the common difference. But this is a longer method, which takes time, so we will use the above method. One should know the ${a_n}$ is the $n$th term in the geometric progression.