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

Hint: 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.