Question

Find the \[{{10}^{th}}\] term of an A.P \[-1.0,-1.5,-2.0,....\]

Answer 1

HInt: Let us assume the first term of a given A.P \[-1.0,-1.5,-2.0,....\] is equal to a. Now let us assume this as equation as equation (1). Let us assume the second term of a given A.P \[-1.0,-1.5,-2.0,....\] is equal to b. Now let us assume this as equation as equation (2). We know that the difference between any consecutive terms gives us the common difference. Now by using equation (1) and equation (2), we can find the common difference. Let us assume this as equation (3). To find the \[{{10}^{th}}\] term of an A.P \[-1.0,-1.5,-2.0,....\] we should assume the value of n is equal to 10. Let us assume this as equation (4). We know that if a is the first term of an A.P and d is the common difference of the A.P, then \[{{n}^{th}}\] term of an A.P is equal to \[{{t}_{n}}\] where \[{{t}_{n}}=a+(n-1)d\]. Now by using the above concept, we can find the \[{{10}^{th}}\] term of an A.P \[-1.0,-1.5,-2.0,....\].

Complete step-by-step answer:
We know that if a is the first term of an A.P and d is the common difference of the A.P, then \[{{n}^{th}}\] term of an A.P is equal to \[{{t}_{n}}\] where \[{{t}_{n}}=a+(n-1)d\].
From the question, we were given an A.P \[-1.0,-1.5,-2.0,....\].
From this question, it is clear that -1 is the first term of the given A.P \[-1.0,-1.5,-2.0,....\].
Let us assume the first term of the given A.P \[-1.0,-1.5,-2.0,....\] is equal to a.
\[\Rightarrow a=-1......(1)\]
Now we have to find the common difference of the given A.P \[-1.0,-1.5,-2.0,....\].
We know that the difference between any consecutive terms gives us the common difference.
Let us assume the second term of an A.P is equal to b.
\[\Rightarrow b=-1.5.......(2)\]
Let us assume the common difference of the given A.P \[-1.0,-1.5,-2.0,....\] is equal to d.
Now from equation (1) and equation (2), we get
\[\begin{align}
  & \Rightarrow d=b-a \\
 & \Rightarrow d=-1.5-(-1) \\
 & \Rightarrow d=-1.5+1 \\
 & \Rightarrow d=-0.5......(3) \\
\end{align}\]
Now from equation (3), it is clear that the common difference of an A.P \[-1.0,-1.5,-2.0,....\] is equal to -0.5.
We know that if a is the first term of an A.P and d is the common difference of the A.P, then \[{{n}^{th}}\] term of an A.P is equal to \[{{t}_{n}}\] where \[{{t}_{n}}=a+(n-1)d\].
Now we have to find the \[{{10}^{th}}\] term of an A.P \[-1.0,-1.5,-2.0,....\]. So, the value of n is equal to 10.
\[\Rightarrow n=10......(4)\]
So, from equation (1), equation (3) and equation (4), we get
\[\begin{align}
  & \Rightarrow {{t}_{10}}=-1+(10-1)(-0.5) \\
 & \Rightarrow {{t}_{10}}=-1+(9)(-0.5) \\
 & \Rightarrow {{t}_{10}}=-1-4.5 \\
 & \Rightarrow {{t}_{10}}=-5.5......(5) \\
\end{align}\]
So, from equation (5) it is clear that the \[{{10}^{th}}\] term is equal to -5.5.

Note: Some students have a misconception that if a is the first term of an A.P and d is the common difference of the A.P, then \[{{n}^{th}}\] term of an A.P is equal to \[{{t}_{n}}=\dfrac{n}{2}[2a+(n-1)d]\]. If this misconception is followed, then the final answer may get interrupted. So, students should have a clear view of this concept. Students should be careful at calculation part of this question. If a small mistake is made, then it is not possible to get a correct final answer. So, one should do the calculation part in a perfect manner. This is also important to solve this problem,