Question

The arithmetic sequences 3, 8, 13, 18,... and 2, 5, 8, 11,... have infinitely many terms in common. Calculate the sum of the first four common terms.

Answer 1

The first four common terms of the arithmetic sequences 3, 8, 13, 18,... and 2, 5, 8, 11,... are 8, 23, 38, and 53. The sum of these common terms is 122. This was determined by using the arithmetic sequence formulas to find common values and summing the first four of them.

Step-by-step explanation:

The arithmetic sequences provided are 3, 8, 13, 18, ... (with a common difference of 5) and 2, 5, 8, 11, ... (with a common difference of 3). To find the common terms, we must find values that are present in both sequences. We know that both sequences can be expressed in a general form. For the first sequence, it is an = 3 + 5(n-1), and for the second, it is bn = 2 + 3(n-1).

To find common terms, we set them equal to each other and solve for n:

3 + 5(n-1) = 2 + 3(n-1)

3 + 5n - 5 = 2 + 3n - 3

5n - 3n = 2 - 3 + 5

2n = 4

n = 2

The second term of the sequences are the same, which are 8. Since both sequences progress in their respective arithmetic paces (with 5 and 3 as the common differences), the next common term will be after a certain number of their respective terms. We need to find a multiple of the common differences (5 and 3) that, when added to the known common term (8), will yield another common term. This happens every 15 steps because 15 is the least common multiple (LCM) of 5 and 3. So, we add 15 repeatedly to each subsequent common term to find the next.

The first four common terms are: 8, 8+15, 8+2(15), 8+3(15), which are 8, 23, 38, and 53 respectively. We then find their sum:

8 + 23 + 38 + 53 = 122

Therefore, the sum of the first four common terms of the given sequences is 122.

Answer 2

The sum of the first four common terms of the given arithmetic sequences is 102.

Step-by-step explanation:

To find the common terms of the two arithmetic sequences 3, 8, 13, 18,... and 2, 5, 8, 11,..., we need to determine a general form for each sequence and then find their intersection.

The first sequence can be written as 3 + 5(n - 1) for the n-th term, and the second sequence can be written as 2 + 3(n - 1) for the n-th term. The common terms will be those where the value of the n-th term in both sequences is the same.

To find the common terms, we can set the two n-th term expressions equal to each other and solve for n. This gives us the equation 3 + 5(n - 1) = 2 + 3(n - 1). Solving for n yields n = 1, 4, 7, ... which are the positions of the common terms in both sequences. The common terms themselves are obtained by substituting these values of n back into either sequence's formula. The first few common terms are 3, 18, 33, ...

The sum of the first four common terms can be found by adding the first four terms: 3 + 18 + 33 + 48 = 102.