Question

Which term of the A.P. $14,9,4, - 1, - 6,..$ is $ - 41$?

Answer 1

Hint: An arithmetic sequence or arithmetic progression is defined as a mathematical sequence in which the difference between two consecutive terms is always a constant. An arithmetic progression is abbreviated as A.P. We also need to learn the three important terms, which are as follows.
$>$ A common difference $\left( d \right)$ is a difference between the two terms.
$>$ ${n^{th}}$term\[({a_n})\]
$>$ And, Sum of the first $n$ terms\[({S_n})\]
Here, we are asked to calculate the ${n^{th}}$ term of the given arithmetic progression.
And the given arithmetic progression is$14,9,4, - 1, - 6,..$
Also, ${n^{th}}$ the term of the given arithmetic progression is$ - 41$.
Formula used:
The formula to calculate the ${n^{th}}$ term of the given arithmetic progression is as follows.
\[\begin{array}{*{20}{l}}
{{{\mathbf{a}}_{\mathbf{n}}} = {\mathbf{a}} + \left( {{\mathbf{n}} - {\mathbf{1}}} \right) \times {\mathbf{d}}}
\end{array}\]
Where, $a$ denotes the first term, $d$ denotes the common difference,$n$is the number of terms, and ${a_n}$ is the ${n^{th}}$ term of the given arithmetic progression.

Complete step-by-step solution:
The given A.P. is $14,9,4, - 1, - 6,..$
Also, it is given that
${a_n} = - 41$
Here, \[a = 14\]
\[d = 9 - 14 = - 5\]
To find: $n$
Now, need to use the formula,
\[\begin{array}{*{20}{l}}
{{{\mathbf{a}}_{\mathbf{n}}} = {\mathbf{a}} + \left( {{\mathbf{n}} - {\mathbf{1}}} \right) \times {\mathbf{d}}}
\end{array}\]
Substitute the known values on the given formula, we get
\[ - 41 = 14 + (n - 1) \times ( - 5)\]
\[ \Rightarrow - 41 = 14 - 5n + 5\]
\[ \Rightarrow - 41 = 19 - 5n\]
\[ \Rightarrow - 41 - 19 = - 5n\]
On solving it further, we have
\[ \Rightarrow - 60 = - 5n\]
\[ \Rightarrow n = 12\]
Hence, the ${12^{th}}$term of the given arithmetic progression is$ - 41$ is the required answer too.

Note: If in the problem it is not mentioned that the given sequence is an A.P or G.P then we will have to find the difference b/w consecutive terms and ratio as well if we get equal difference then the given sequence would be an A.P otherwise if we get the same ratio then the given sequence is G.P but here in the problem here already given that the given sequence is an A.P so we use the formulae of an A.P. Common difference=$\left( d \right)$ is the difference between the two terms.
The ${n^{th}}$ term of the given arithmetic progression.
And the given arithmetic progression is $14,9,4, - 1, - 6,..$
Hence, the ${12^{th}}$ term of the given arithmetic progression i s$ - 41$.