Two series are 16, 21, 26, ... and 17, 21, 25, .... What is the sum of the…

When two arithmetic progressions (APs) share infinitely many common values, those common values themselves form a new AP: its first term is the smallest number that appears in both original series, and its common difference equals the LCM of the two original series' common differences. The sum of the first n terms of this new AP follows the standard formula S = (n/2)[2a + (n-1)d].

1. Series A (16, 21, 26, ...) has a common difference of 5, and Series B (17, 21, 25, ...) has a common difference of 4.

2. The LCM of the two common differences, LCM(5, 4) = 20, so once a value is common to both series, the next common value appears exactly 20 later.

3. The first term common to both series is 21 (it is the second term of Series A: 16 + 5 = 21, and the second term of Series B: 17 + 4 = 21).

4. So the common terms form a new AP: 21, 41, 61, ..., with first term a = 21, common difference d = 20, and we need the sum of n = 100 such terms.

5. Applying the AP sum formula: S = (n/2)[2a + (n-1)d] = (100/2)[2*21 + 99*20] = 50 * [42 + 1980] = 50 * 2022 = 101100.

Cross-check using the last-term method: the 100th common term is a + (n-1)d = 21 + 99*20 = 2001, so S = n*(a + last term)/2 = 100*(21 + 2001)/2 = 100*2022/2 = 101100 - the same value, confirming the sum.

So the sum of the first hundred common numbers is 101100.