Question

Which term of the A.P 25, 20, 15… is the first negative term.

Answer 1

Hint: We will use the formula of arithmetic progression i.e. \[{a_n} = a + (n - 1) \times d\] and put the values of a and d in the formula and we will find the possible value of n for which \[{a_n} < 0\] . Then, we will put the value of n in the formula of arithmetic progression. On simplification, we will get the answer.

Complete step-by-step answer:
Given the series,
25, 20, 15, …
So, the first term of this series is 25 i.e. a = 25 …. (1)
The common difference is \[ - 5\] i.e. d = - 5 … (2)
Now, we have to find the first negative term. So, we will use the formula
\[ \Rightarrow \]\[{a_n} = a + (n - 1) \times d\] … (3)
We have to find the first negative term. Let the term be \[{a_n}\] .
i.e. \[{a_n} < 0\] … (4)
Putting the value of equation (3) in (4), we get
\[ \Rightarrow \]\[a + (n - 1)d < 0\] … (5)
Substitute the value of a and d from equation (1) and (2) to equation (5).
\[ \Rightarrow \]\[25 + (n - 1) \times ( - 5) < 0\]
Taking \[(n - 1) \times ( - 5)\] to the R.H.S., we get
\[ \Rightarrow \]\[25 < 5 \times (n - 1)\] … (6)
Dividing equation (6) by 5, we get
\[ \Rightarrow \]\[\dfrac{{25}}{5} < \dfrac{{5 \times (n - 1)}}{5}\]
On simplification, we get
\[ \Rightarrow \]\[5 < n - 1\] … (7)
Adding 1 on both sides, we get
\[ \Rightarrow \]\[5 + 1 < n\]
Adding 5 and 1, we get
\[ \Rightarrow \]\[6 < n\]
i.e.\[n > 6\]
\[ \Rightarrow \]\[n = 7\]… (8)
Putting the value of equation (8) in equation (3)
\[ \Rightarrow \]\[{a_7} = 25 + (7 - 1) \times ( - 5)\]
On simplification, we get
\[ \Rightarrow \]\[{a_7} = 25 + 6 \times ( - 5)\]
Multiplying 6 and -5, we get
\[ \Rightarrow \]\[{a_7} = 25 + ( - 30)\]
On simplification, we get
\[ \Rightarrow \]\[{a_7} = 25 - 30\]
So, we have
\[ \Rightarrow \]\[{a_7} = - 5\]
Hence, the seventh term of the series is -5 and it is the first negative term.

Note: Arithmetic progression is the sequence of numbers in which each number differs from the preceding one by a constant quantity. An example of arithmetic progression is 1, 4, 7,10, 13… because each term differs from the preceding one by 3.