Question

Given term 3, 7, 11, 15, 19, …….. are in AP. Find the 25th term.
A.96
B.99
C.102
D.106

Answer 1

Hint: Here we will first find the common difference of the given series. Then we will use the formula of the \[{n^{th}}\] term of an arithmetic progression and substitute the value of the first term and the common difference in the formula to get the required term of the given arithmetic progression.

Formula used:
\[{n^{th}}\]term of an arithmetic progression is given by \[{a_n} = a + \left( {n - 1} \right) \times d\], where \[a\] is the first term of an AP, \[d\] is the common difference and \[{a_n}\] is the \[{n^{th}}\] term of an arithmetic progression.

Complete step-by-step answer:
Here we need to find the 25th term of the given sequence which is in AP.
The given AP is 3, 7, 11, 15, 19, ……..
We can see that the first term of the given AP is equal to 3 i.e. \[a = 3\].
Now, we will calculate the common difference of the AP which we will find by calculating the difference between any two consecutive numbers.
\[d = 7 - 3 = 11 - 7 = 4\]
We need to calculate the value of the 25th term i.e. \[n = 25\]
Now, substituting \[a = 3\], \[d = 4\] and \[n = 25\] in the formula \[{a_n} = a + \left( {n - 1} \right) \times d\], we get
\[{a_{25}} = 3 + \left( {25 - 1} \right) \times 4\]
Subtracting the terms inside the bracket, we get
\[ \Rightarrow {a_{25}} = 3 + 24 \times 4\]
Multiplying the terms, we get
\[ \Rightarrow {a_{25}} = 3 + 96\]
Now, we will add these two numbers. Therefore, we get
\[ \Rightarrow {a_{25}} = 99\]
Therefore, the 25th term of the given AP is equal to 99.
Hence, the correct option is option B.

Note: Here AP stands for Arithmetic Progression and it is defined as the sequence of the numbers which are in order and in which the difference of any two consecutive numbers is a constant value. There are other types of progression also such as geometric and harmonic progression. Geometric progression is a sequence or series in which the two consecutive numbers have a common ratio. However, harmonic progression is a reciprocal of an arithmetic progression.