Question

Find the number of terms in the given AP, \[7,13,19.......205\]

Answer 1

Hint: A sequence is known as A.P when the difference between any two consecutive common terms is the same.
In an arithmetic program, the first term of the sequence is ‘a’ and the common difference is ‘d’. And the formula of nth terms is given by\[{a_n} = a + (n - 1)d\], where n is the number of terms in the sequence.
In the given series value of \[a\,is\,7\] and \[d = 6\]
Therefore,

Complete step by step answer:
Given \[7,13,19,.........205\]
Here first term of the sequence is \[a = 7\]
Common difference is \[d = 13 - 7 = 6\]
Last term of the sequence= \[{a_n} = 205\]
Now to find the number of terms in the sequence we have:
\[an = a + (n - 1)d\]
On substituting a= 7, d=6 and \[{a_n} = 205\], we get:
\[ \Rightarrow 205 = 7 + (n - 1)6\]
\[ \Rightarrow 205 - 7 = (n - 1)6\]
\[ \Rightarrow 198 = (n - 1)6\]
\[ \Rightarrow \dfrac{{198}}{6} = n - 1\]
\[ \Rightarrow 33 + 1 = n\]
\[ \Rightarrow n = 34\]
So, the number of terms in the sequence are 34.

Note:
In an A.P we can also find \[{n^{th}}\] term from the end by using formula \[l - (n - 1)\] d where l is the last term of the sequence and n is the number of terms of the sequence.
We must never get confused in recognizing the A.P because in A.P common difference is the same throughout the sequence.