Question

How do you find the 15th term in the arithmetic sequence  $  - 3,4,11,18,...? $ ?

Answer 1

Hint: The general sequence of Arithmetic progression(A.P.) is $a, a+d, a+2d, a+3d, .....$. Here $a$ is the first term  and $d$ is the difference between any two consecutive terms, called the common difference.

$n^{th}$ term of an A.P. is given by $a_n= a+(n-1)d$.

In order to determine the 15th term of the given arithmetic sequence, we relate the given numbers with the general sequence of A.P. and Using the $n^{th}$ term formula, we find the 15th term in the given A.P.

Complete step-by-step answer:

The given sequence is an Arithmetic Progression (A.P.) .

In our sequence first term  $ a =  - 3 $ 

Common difference (d) can be calculated by subtracting any two consecutive terms, we get  $ d = 4 - \left( { - 3} \right) = 4 + 3 = 7 $ .

As we know the nth term of an A.P. is  $ {a_n} = a + (n - 1)d $ 

where $a$ is the first term, $d$ is the common constant difference

According to the question we have to find the value of the 15th term, so  $ n = 15 $ 

So the 15th term of the A.P. will be  $ {a_{15}} $ 

Now putting the values of  $ n,a\,and\,d $ in the nth term of A.P. we get

 $   {a_n} = a + (n - 1)d $

Substituting the corresponding values,

 $ {a_{15}} =  - 3 + (15 - 1)\left( 7 \right)$

On simplification,

 $ {a_{15}} =  - 3 + (14)\left( 7 \right) $

 $\Rightarrow {a_{15}} =  - 3 + (14)\left( 7 \right) $

On further simplification,

 $ {a_{15}} =  - 3 + 98 $

 $ {a_{15}} = 95 $ 

Therefore, the 15th term $ \left( {{a_{15}}} \right) $ of the given arithmetic sequence is equal to  $ 95 $.

Additional information:

In order to determine whether a sequence is an A.P. or not when its nth term is given, we may use the following algorithm .

Algorithm:

Step 1: Obtain  $ {a_n} $ .

Step 2: Replace  $ n $ by $ n + 1 $ in  $ {a_n} $ to get  $ {a_n} + 1 $ 

Step 3: Calculate  $ {a_{n + 1}} - {a_n} $ .

Step 4: If   $ {a_{n + 1}} - {a_n} $ is independent of n , the given sequence is an A.P. Otherwise is not an A.P.  

Note: 1. Sequence: A sequence is a function whose domains is the set of N of natural numbers.
2. Real Sequence: A sequence whose range is a subset of R is called a real sequence.
In other words, a real sequence is a function having domain N and range equal to a subset of the set R of real numbers.
3. Arithmetic Progression (A.P): A sequence is called an arithmetic progression if the difference of a term and the previous term is always the same.
i.e. $ {a_{n + 1}} - {a_n} = $ constant $ ( = d) $ for all $ n \in N $ .
The constant difference is generally denoted by $d$ which is called as the common difference.