Question

Find the number of terms in each of the following AP’s
\[7,13,19,....205\]

Answer 1

Hint: Arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.
Difference here means the second minus the first term.
If the initial term of an arithmetic progression is \[{a_1}\]and the common difference of successive members is d them the \[{n^{th}}\] term of the sequence \[{a_n}\] is given by
\[{a_n} = {a_1} + (n - 1)d\]
The behaviour of the AP depends on the common difference d. If the common difference is
\[ \to \]Positive, then the terms will grow towards positive infinity.
\[ \to \] Negative, then the term will grow towards negative infinity.
Now, In the formula
\[{a_n} = {a_1} + (n - 1)d\]
\[ \to {a_1}\] is the first term of an AP
\[ \to \] \[{a_n}\] is the \[{n^{th}}\] term of an AP
\[ \to \] d is the difference between terms of the AP
\[ \to \] n is the number of terms in the AP.

Complete step-by-step answer:
Given AP is
\[7,13,19,....205\]
Using the formula
\[{a_n} = a + (n - 1)d\]
Here, first term of AP is
 \[a = 7\]
To find the value of $d$ we need to subtract the second term from first term.
\[d = (Second\,term - First\,term)\]
\[d = 13 - 7 = 6\]
Last term is \[an = 205\]
Now, find A.P.
\[{a_n} = a(n - 1)d\]
Put values of each,
We get,
\[\begin{gathered}
  205 = 7 + (n - 1) \times 6 \\
  205 - 7 = (n - 1) \times 6 \\
  198 = (n - 1) \times 6 \\
\end{gathered} \]
Simplify
\[\begin{gathered}
  \dfrac{{198}}{6} = n - 1 \\
  33 = n - 1 \\
  33 + 1 = n \\
  34 = n \\
\end{gathered} \]
We have,
$n = 34$
Hence there are 34 terms in the given AP.

Note: The sum of the members of finite AP is called on arithmetic series.
For example, consider the sum:
\[2 + 5 + 8 + 11 + 14\]
This sum can be found quickly by taking the number ‘n’ of term bring added (here 5), Multiplying by the sum of the first and last number in the progression and dividing by 2.
\[\begin{gathered}
  \dfrac{{n({a_1} + {a_n})}}{2}\,\,\, \\
  n = 5 \\
  {a_1} = 2 \\
  an = 14 \\
\end{gathered} \]
In the case above, this gives the equation
\[2 + 5 + 8 + 11 + 14 = \dfrac{{5(2 + 14)}}{2} = 40\]