Question

How to find the first term, the common difference, and the $n$th term of the arithmetic sequence here? 8th term is 4; 18th term is -96?

Answer 1

We just need to know the general formula of the arithmetic sequence. Since two of the terms are given we can apply them in the formula of the arithmetic sequence to get two different simultaneous equations having two unknown variables and can solve them to find those variables.

Complete step-by-step answer:
Let’s take the general terminology and symbols for the arithmetic sequence. That is let $a$ be the first term and $d$ be the common difference. So the formula for the $n$th term of an arithmetic sequence becomes:
$\bullet$ ${T_n} = a + \left( {n - 1} \right)d$
Now according to the question the 8th term is $4$and the 18th term is $- 96$
By substituting the values in the above formula we get:
For 8th term
$\bullet$ $a + \left( {8 - 1} \right)d = 4$
$\bullet$ $a + 7d = 4$ ; (equation1)
For 18th term
$\bullet$ $a + \left( {18 - 1} \right)d = - 96$
$\Rightarrow a + 17d = - 96$ ; (equation2)
Since the term $a$ is common in both the equation1 and equation 2 we can subtract them and get the result for the common difference $d$
By subtracting equation1 form equation2 we get:
$\bullet$ $a + 17d - \left( {a + 7d} \right) = - 96 - \left( 4 \right)$
$\Rightarrow a + 17d - a - 7d = - 100$
$\Rightarrow 10d = - 100$
$\Rightarrow d = - 10$
Now we have the common difference $d = - 10$, which mean that this arithmetic sequence is a decreasing type. So now we can get the value of the first term $a$ by substituting the value of $d$ in any of the equation.
Let’s substitute $d = - 10$ in equation 1:
$\bullet$ $a + \left( {8 - 1} \right)d = 4$
$\Rightarrow a + 7 \times \left( { - 10} \right) = 4$
$\Rightarrow a - 70 = 4$
$\Rightarrow a = 74$
Now we got $a = 74$ so the first term is $74$
Now we need to find the $n$th term of an arithmetic sequence:
So according to the formula:
$\bullet$ ${T_n} = a + \left( {n - 1} \right)d$
Now substituting $a = 74$ for the first term and for the common difference $d = - 10$we get:
$\bullet$ ${T_n} = 74 + \left( {n - 1} \right) \times \left( { - 10} \right)$
Or we can also write it as
$\bullet$ ${T_n} = 74 - 10\left( {n - 1} \right)$
So we have the first term $= 74$, the common difference $= - 10$and the $n$th term of an arithmetic sequence as ${T_n} = 74 - 10\left( {n - 1} \right)$.

Note: Start simply by using the general formula of the sequence whether it's arithmetic or geometric, since these kinds of questions generally come with two different terms of the sequence and ask us to find the two unknown parameters of the sequence. We will eventually end up with two different equations having two unknown variables. The major mistake in these types of questions occurs only when solving the simultaneous equation.