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Answer :
Final answer:
The difference between the sums of the first 50 terms of two APs with the same common difference but different first terms is 250.
Explanation:
To find the difference between the sums of the first 50 terms of two arithmetic progressions (APs) with the same common difference and different first terms, we use the formula for the sum of the first n terms of an AP: Sn = n/2 (2a + (n-1)d), where Sn is the sum of the first n terms, a is the first term, d is the common difference, and n is the number of terms.
For the first AP with a=3, the sum of the first 50 terms S50 is 50/2 (2*3 + (50-1)*d) = 25 (6 + 49d).
For the second AP with a=8, the sum of the first 50 terms S50 is 50/2 (2*8 + (50-1)*d) = 25 (16 + 49d).
The difference between the two sums is 25 (16 + 49d) - 25 (6 + 49d) = 25(16-6) = 25*10 = 250.
Thus, the difference between the sums of their first 50 terms is 250.
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