Question

Which term of the AP 121, 117, 113, …. has the first negative term?

Answer 1

Hint: Find the first term and the common difference of the AP. Then use the formula for the nth term of the AP which is \[{t_n} = a + (n - 1)d\] and find the first negative term.

Complete step-by-step answer:
Arithmetic Progression is a sequence of numbers in which each differs from the preceding one by a constant quantity. This constant quantity is called the common difference.
The given AP is given as follows:
121, 117, 113,……
The first term of the AP is represented by the letter ‘a’ and the first term of the above AP is 121. The common difference is represented by the letter ‘d’ and is the difference between any two consecutive terms of the AP. Hence, we have as follows:
\[a = 121..........(1)\]
\[d = 117 - 121\]
\[d = - 4...........(2)\]
The first term and the common difference completely describe an AP. We can find all the terms in the AP using these two quantities. The formula to calculate the nth term of the AP is given as follows:
\[{t_n} = a + (n - 1)d\]
For the first negative term, we need to find the smallest n such that \[{t_n} < 0\].
Substituting equation (1) and equation (2) in the above equation, we have:
\[121 + (n - 1)( - 4) < 0\]
Simplifying the expression, we have:
\[121 - 4n + 4 < 0\]
Taking n to the right-hand side of the inequality, we have:
\[125 < 4n\]
Solving for n, we have:
\[n > \dfrac{{125}}{4}\]
\[n > 31.25\]
Hence, the smallest integer value of n is 32.
The given AP has the 32nd term as the first negative term.

Note: We may get the 31st term as the answer, where the 31st term of the AP is zero but the question is asked to find the first negative term. Hence, the 31st term is a wrong answer, zero is not a negative term.