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The smallest number of terms of the AP that will make the sum of terms positive is 73.
Since we need to know the number for the sum of terms, we find the sum of terms of the AP
The sum of terms of an AP is given by S = n/2[2a + (n - 1)d] where
Since we have the AP "-54,-52.5,-51,-49.5" ....", the first term, a = -54 and the second term, a₂ = -52.5.
The common difference, d = a₂ - a
= -52.5 - (-54)
= -52.5 + 54
= 1.5
Since we require the sum of terms , S to be positive for a given number of terms, n.
So, S ≥ 0
n/2[2a + (n - 1)d] ≥ 0
So, substituting the values of the variables into the equation, we have
n/2[2(-54) + (n - 1) × 1.5] ≥ 0
n/2[-108 + 1.5n - 1.5] ≥ 0
n/2[1.5n - 109.5] ≥ 0
n[1.5n - 109.5] ≥ 0
So, n ≥ 0 or 1.5n - 109.5 ≥ 0
n ≥ 0 or 1.5n ≥ 109.5
n ≥ 0 or n ≥ 109.5/1.5
n ≥ 0 or n ≥ 73
Since n > 0, the minimum value of n is 73.
So, the smallest number of terms of the AP that will make the sum of terms positive is 73.
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