Question

What is the \[{12^{th}}\] term of the sequence \[5,12,19,...\]?

Answer 1

Hint: In this question, we have to find out the required value from the given particulars.
We need to first find out the common difference & the first term. By subtracting the first term from the second term, we will get the common difference. Then putting all the values and the number of terms in the formula of the nth term of the arithmetic sequence, we can find out the required solution.
Property of A.P.:
The nth term of the arithmetic sequence is
\[{a_n} = a + \left( {n - 1} \right)d\]
Where,
a = first term of the sequence
d = common difference
n= number of terms

Complete step-by-step solution:
It is given that the sequence \[5,12,19,...\].
We need to find the \[{12^{th}}\] term of the sequence \[5,12,19,...\].
a = the first term of the sequence =\[5\].
d = the common difference = second term – first term =\[12 - 5 = 7\] which is also the common difference between third and second term.
Thus \[5,12,19,...\] is an arithmetic sequence.
Hence, we can apply the formula of the nth term of the arithmetic sequence, which is
\[{a_n} = a + \left( {n - 1} \right)d\]
Where,
a = first term of the sequence
d = common difference
n= number of terms
Here, a = \[5\]\[d = 7\]& n = number of terms =\[10\].
Therefore, the \[{12^{th}}\] term of the sequence \[5,12,19,...\]. is
\[{a_{12}} = 5 + \left( {12 - 1} \right) \times 7\]
\[\Rightarrow {a_{12}} = 5 + 11 \times 7\]
\[\Rightarrow {a_{12}} = 5 + 77\]
\[\Rightarrow {a_{12}} = 82\]
Hence, the \[{12^{th}}\] term of the sequence \[5,12,19,...\] is \[82\] .

Note: An arithmetic progression is a sequence of numbers such that the difference of any two successive members is a constant. In General, we write an Arithmetic Sequence like this:\[\left\{ {a,a + d,a + 2d,a + 3d....} \right\}\], Where a is the first term, and d is the difference between the terms
(Called the “common difference”).