Which term of the A.P. 32, 28, 24, 20, ... is the first negative term?
(1) 9th
(2) 8th
(3) 10th
(4) 11th
Answer (2)
To find which term of the arithmetic progression (A.P.) 32, 28, 24, 20, ... is the first negative term, we first need to understand the pattern of the sequence.
Identify the first term and common difference:
The first term a 1 is 32.
The common difference d can be found by subtracting the second term from the first term: 28 − 32 = − 4 .
General formula for the n -th term of an A.P.:
The n -th term a n of an arithmetic progression is given by the formula: a n = a 1 + ( n − 1 ) × d
Substituting the known values: a n = 32 + ( n − 1 ) × ( − 4 ) a n = 32 − 4 n + 4 a n = 36 − 4 n
Find the first negative term:
We want a n < 0 .
Solving the inequality: 36 − 4 n < 0 36 < 4 n 9 < n
This tells us that n must be greater than 9.
Therefore, the 10th term is the first negative term.
Let's verify by calculating the 10th term:
a 10 = 36 − 4 × 10 a 10 = 36 − 40 a 10 = − 4
The 10th term is indeed negative, confirming our solution.
Therefore, the correct answer is (3) 10th .
The first negative term of the arithmetic progression 32, 28, 24, 20, ... is the 10th term. This is determined by using the formula for the n -th term and solving for when it becomes negative. Thus, the answer is (3) 10th .
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