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

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

Explanation

Solution

We will use the formula of arithmetic progression i.e. an=a+(n−1)×d{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 an<0{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 - 5 i.e. d = - 5 … (2)
Now, we have to find the first negative term. So, we will use the formula
\Rightarrow $$$${a_n} = a + (n - 1) \times d … (3)
We have to find the first negative term. Let the term be an{a_n} .
i.e. an<0{a_n} < 0 … (4)
Putting the value of equation (3) in (4), we get
\Rightarrow $$$$a + (n - 1)d < 0 … (5)
Substitute the value of a and d from equation (1) and (2) to equation (5).
\Rightarrow $$$$25 + (n - 1) \times ( - 5) < 0
Taking (n−1)×(−5)(n - 1) \times ( - 5) to the R.H.S., we get
\Rightarrow $$$$25 < 5 \times (n - 1) … (6)
Dividing equation (6) by 5, we get
\Rightarrow $$$$\dfrac{{25}}{5} < \dfrac{{5 \times (n - 1)}}{5}
On simplification, we get
\Rightarrow $$$$5 < n - 1 … (7)
Adding 1 on both sides, we get
\Rightarrow $$$$5 + 1 < n
Adding 5 and 1, we get
\Rightarrow $$$$6 < n
i.e.n>6n > 6
\Rightarrow $$$$n = 7… (8)
Putting the value of equation (8) in equation (3)
\Rightarrow $$$${a_7} = 25 + (7 - 1) \times ( - 5)
On simplification, we get
\Rightarrow $$$${a_7} = 25 + 6 \times ( - 5)
Multiplying 6 and -5, we get
\Rightarrow $$$${a_7} = 25 + ( - 30)
On simplification, we get
\Rightarrow $$$${a_7} = 25 - 30
So, we have
\Rightarrow $$$${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.