How many positive terms are there in the series: 75,72,69,…,−6,−9,−12?
Explanation
1. Identify the Arithmetic Progression (AP):
The given progression is: 75,72,69,…
2. Set up the Inequality for Positive Terms:
Any term in an AP is represented by the formula:
an=a+(n−1)d
Since we only want to find the positive terms, the value of an must be strictly greater than zero (a_n > 0):
75 + (n - 1)(-3) > 0
3. Solve for n:
75 - 3n + 3 > 0
78 - 3n > 0
78 > 3n
\frac{78}{3} > n
26 > n \implies n < 26
Since the number of terms (n) must be a whole integer, the largest integer value strictly less than 26 is:
n=25
Correct Answer
Option (a) 25