Find the number of terms common to the sequences 14, 17, 20, …149 and 10, 14, 18, …158.10111213

Sure, let's solve this step by step.

Step 1: Identify the common difference in both sequences. In the first sequence, the common difference is 17 - 14 = 3. In the second sequence, the common difference is 14 - 10 = 4.

Step 2: Find the first common term in both sequences. The first common term is 14.

Step 3: Find the last common term in both sequences. The last common term should be less than or equal to 149 (from the first sequence) and 158 (from the second sequence). The last common term is 146.

Step 4: Use the formula for the nth term of an arithmetic sequence to find the number of common terms. The formula is a + (n - 1)d, where a is the first term, d is the common difference, and n is the number of terms. We can rearrange this formula to solve for n: n = [(last term - first term) / common difference] + 1

For the common terms in these sequences, the first term a is 14, the last term is 146, and the common difference d is the least common multiple of 3 and 4, which is 12.

So, n = [(146 - 14) / 12] + 1 = 12. Therefore, there are 12 terms common to both sequences.