Question

Find the 8^th term from the End of the A.P. 7, 10, 13, ..., 184

Answer 1

Hint: Here in this question, given a sequence of arithmetic progression we have to find the value of 8^th term from the end of the sequence. To find this by using a formula $${a_n} = a + (n - 1)d$$, where $${a_n}$$ is the value of nth term of A.P sequence, $$a$$ is the value of first term and $$d$$ is the common difference.

Complete step by step answer:
The general arithmetic progression is of the form $$a,a + d,a + 2d,...$$ where a is first term nth d is the common difference which is the same between the distance on any two numbers in sequence. The nth term of the arithmetic progression is defined as $${a_n} = a + (n - 1)d$$.
Now let us consider the A.P. sequence which is given in the question 7, 10, 13, ..., 184. Let we find the term i.e., $$n$$ where the last number 184 is located.
Here $$a = {a_1} = 7$$, $${a_2} = 10$$, $${a_3} = 13$$, and, $${a_n} = 184$$
The common difference:
$$ \Rightarrow d = {a_2} - {a_1}$$
$$ \Rightarrow d = 10 - 7$$
$$ \Rightarrow d = 3$$
Let we find the $$n$$ value by considering the formula of nth term of the arithmetic progression i.e.,
$$ \Rightarrow \,\,{a_n} = a + (n - 1)d$$
On substituting the values, then we have
$$ \Rightarrow \,\,184 = 7 + (n - 1)3$$
Remove a parenthesis by multiplying a 5.
$$ \Rightarrow \,\,184 = 7 + 3n - 3$$
On simplification, we have
$$ \Rightarrow \,\,184 = 4 + 3n$$
Subtract 4 on both side, then
$$ \Rightarrow \,\,184 - 4 = 3n$$
$$ \Rightarrow \,\,180 = 3n$$
Divide 3 on both side, then we get
$$ \Rightarrow \,\,\dfrac{{180}}{3} = n$$
$$ \Rightarrow \,\,60 = n$$
Or
$$ \Rightarrow \,\,n = 60$$
So, the 8^th term from the end means
$$ \Rightarrow \,\,60 - 8 + 1= 53$$
53^rd term from the beginning.
So, for the 53^rd term $$n = 53$$
$$ \Rightarrow \,\,{a_{53}} = 7 + (53 - 1)3$$
$$ \Rightarrow \,\,{a_{53}} = 7 + (52)3$$
$$ \Rightarrow \,\,{a_{53}} = 7 + 156$$
$$\therefore \,\,{a_{53}} = 163$$
Therefore, the 8^th term from the end of the given A.P. is 163.

Note:
By considering the formula of arithmetic sequence we verify the obtained the value which we
obtained. We have to check the common difference for all the terms. Suppose if we check for first
two terms not for other terms then we may go wrong. So, definition of arithmetic sequence is
important to solve these kinds of problems.