Question

How do I find the \[{n^{th}}\] term of the arithmetic sequence whose initial term and common difference term \[d\] are given below \[?\]
A. \[{a_1} = 4,d = 3\]
B. \[{a_1} = - 8,d = 5\]

Answer 1

Hint: This question describes the operation of addition/ subtraction/ multiplication/ division. In this question, we assume the initial term as \[a\] and the common difference term as \[d\] . We need to know the basic equation to determine the \[{n^{th}}\] term of the arithmetic equation. Also, we need to find two answers for two cases given in the equation.

Complete step-by-step answer:
In this question, we need to find the \[{n^{th}}\] term of the arithmetic sequence. To solve the given question we need to know the basic equation to find \[{n^{th}}\] a term which is given below,
 \[{n^{th}} = a + \left( {n - 1} \right)d \to \left( 1 \right)\]
Here, \[a\] is the initial term of an arithmetic sequence,
 \[d\] is the common difference term of an arithmetic sequence.
In the question, we have two cases which are,
 \[{a_1} = 4,d = 3\]
 \[{a_1} = - 8,d = 5\]
Let’s solve the problem with the case \[\left( A \right)\] .
In case \[\left( A \right)\] we have \[{a_1} = 4,d = 3\]
Let’s substitute the above values in the equation \[\left( 1 \right)\] , we get
 \[\left( 1 \right) \to {n^{th}} = a + \left( {n - 1} \right)d\]
 \[{n^{th}} = 4 + \left( {n - 1} \right) \times 3\]
So, we get
 \[
  {n^{th}} = 4 + 3n - 3 \\
  {n^{th}} = 1 + 3n \;
 \]
In case \[\left( B \right)\] we have \[{a_1} = - 8,d = 5\]
Let’s substitute the above-mentioned values in the equation \[\left( 1 \right)\] , we get
 \[\left( 1 \right) \to {n^{th}} = a + \left( {n - 1} \right)d\]
 \[
  {n^{th}} = - 8 + \left( {n - 1} \right) \times 5 \\
  {n^{th}} = - 8 + 5n - 5 \;
 \]
So, we get
 \[{n^{th}} = - 13 + 5n\]
So, the final answer is
So, the correct answer is “ \[{n^{th}} = - 13 + 5n\] ”.
The \[{n^{th}}\] term of the arithmetic sequence \[{a_1} = 4,d = 3\] is given below,
 \[{n^{th}} = 1 + 3n\]
The \[{n^{th}}\] term of the arithmetic sequence \[{a_1} = - 8,d = 5\] is given below,
 \[{n^{th}} = - 13 + 5n\]
So, the correct answer is FOR A. “ \[{n^{th}} = 1 + 3n\] ”.
So, the correct answer is “ FOR B. “ \[{n^{th}} = - 13 + 5n\] ”.

Note: This question describes the operation of addition/ subtraction/ multiplication/ division. Remember the formula to find the \[{n^{th}}\] term of the arithmetic sequence. Also, note the following things,
When a negative number is multiplied with the negative number the answer becomes a positive number.
When a positive number is multiplied with the positive number the answer becomes a positive number.
When a negative number is multiplied with the positive number the answer becomes a negative number.