Question

Find the \[{19^{th}}\] of A.P \[7,13,19,25...\]

Answer 1

Hint: When a difference between any two terms in a sequence is the same then it is called Arithmetic Progression (A.P). The three main terms in arithmetic progression are Common difference, \[{n^{th}}\] term and sum of first \[n\] terms and they are denoted by \[d,{a_n}\] and \[{S_n}\] respectively.
Let \[{a_1},{a_2},{a_3},...,{a_n}\] be the arithmetic progression then, common difference (d) is defined as \[d = {a_2} - {a_1}\] .
Thus, the arithmetic progression can also be written as \[a,a + d,a + 2d,...,a + (n - 1)d\] .
To find a particular term in the arithmetic progression, we have to apply a formula that is, \[{a_n} = a + (n - 1)d\] . This will yield the \[{n^{th}}\] term.

Complete step by step answer:
The given arithmetic progression is \[7,13,19,25...\]
Since they have directly said that this is the arithmetic progression, we can directly proceed to find the solution. Otherwise, we have to find whether this is an A.P.
The first term in the given arithmetic progression is \[a = 7\] .
Let us find the common difference in this arithmetic progression.
The formula to find the common difference is, \[d = {a_2} - {a_1}\]
So, \[d = 13 - 7\]
  \[ \Rightarrow d = 6\]
Thus, the common difference of this arithmetic progression is \[d = 6\] .
Now we have all the things that we need to find a particular term in this arithmetic progression.
 \[a = 7\] And \[d = 6\] now let’s find the \[{19^{th}}\] term in this arithmetic progression.
Formula to find the \[{n^{th}}\] term in the arithmetic progression is \[{a_n} = a + (n - 1)d\] .
Here we have to find the \[{19^{th}}\] term so \[n = 19\] .
Substituting the above terms in the formula we get,
 \[{a_{19}} = 7 + (19 - 1)6\]
On simplifying this we get,
 \[ = 7 + (18)6\]
 \[ = 7 + 108\]
 \[{a_{19}} = 115\]
Thus the \[{19^{th}}\] term in the given arithmetic progression is \[115\]

Note: In this problem they have directly said that this is the arithmetic progression, we can directly proceed to find the solution. Otherwise, we have to find whether this is an A.P. We know that, in arithmetic progression the common difference between any two numbers is always the same; it is also the same when the sequence has odd or even number of terms.