What is the common difference for this arithmetic sequence? 31,48,65,82,…
The Correct answer and Explanation is:
To find the common difference in an arithmetic sequence, you simply subtract any term from the subsequent term. The arithmetic sequence given is:
31, 48, 65, 82, …
Let’s denote the first term as a1a_1a1, the second term as a2a_2a2, the third term as a3a_3a3, and the fourth term as a4a_4a4:
- a1=31a_1 = 31a1=31
- a2=48a_2 = 48a2=48
- a3=65a_3 = 65a3=65
- a4=82a_4 = 82a4=82
The common difference ddd can be calculated as follows:
- From the first and second terms:d=a2−a1=48−31=17d = a_2 – a_1 = 48 – 31 = 17d=a2−a1=48−31=17
- From the second and third terms:d=a3−a2=65−48=17d = a_3 – a_2 = 65 – 48 = 17d=a3−a2=65−48=17
- From the third and fourth terms:d=a4−a3=82−65=17d = a_4 – a_3 = 82 – 65 = 17d=a4−a3=82−65=17
Each calculation yields a common difference of 17, confirming that the sequence is indeed arithmetic.
Explanation of Arithmetic Sequences
An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is referred to as the common difference (ddd).
The formula for the nnn-th term of an arithmetic sequence can be expressed as:an=a1+(n−1)da_n = a_1 + (n – 1)dan=a1+(n−1)d
In this case, we see that d=17d = 17d=17. This means that each term is obtained by adding 17 to the previous term. For instance:
- Starting with 31 (the first term), adding 17 gives us the second term (48).
- Adding 17 to 48 gives us the third term (65).
- Adding 17 to 65 gives us the fourth term (82).
The constant addition of 17 to each term illustrates the defining feature of arithmetic sequences, allowing us to easily predict future terms in the sequence. Thus, the common difference of this arithmetic sequence is 17.