Final answer:
The first term (a_1) of the arithmetic sequence is 2 and the common difference (d) is 4. This is found by setting up two equations from the given 4th and 11th terms and solving for the two unknowns, a_1 and d.
Step-by-step explanation:
The student is asking how to find the first term and the common difference of an arithmetic sequence when the 4th term is 14 and the 11th term is 42. In an arithmetic sequence, each term is equal to the previous term plus a constant difference, which is referred to as the common difference (d). The general form of an arithmetic sequence can be expressed as:
a_n = a_1 + (n - 1)d,
where a_n is the nth term, a_1 is the first term, and d is the common difference. We can set up two equations based on the given information:
- 14 = a_1 + 3d (since the 4th term is 14)
- 42 = a_1 + 10d (since the 11th term is 42)
By subtracting the first equation from the second, we can find the common difference:
28 = 7d ⇒ d = 4.
Now that we know the common difference, we can substitute it back into either of the original equations to solve for the first term:
14 = a_1 + 3(4) ⇒ 14 = a_1 + 12 ⇒ a_1 = 2.
Therefore, the first term of the sequence is 2, and the common difference is 4.