The first 3 terms of a sequence are given. Write your answer as a decimal or hole number round to the nearest thousand twelve seventeen twenty two, find the thirty six term
The correct answer and explanation is:
Let’s analyze the sequence given:
Sequence: 12, 17, 22, …
Step 1: Identify the pattern
Look at the differences between consecutive terms:
- 17−12=517 – 12 = 5
- 22−17=522 – 17 = 5
The difference between each term is constant (+5), so this is an arithmetic sequence with:
- First term a1=12a_1 = 12
- Common difference d=5d = 5
Step 2: Find the 36th term formula
The nn-th term of an arithmetic sequence is given by: an=a1+(n−1)da_n = a_1 + (n – 1)d
Step 3: Calculate the 36th term
Substitute a1=12a_1 = 12, d=5d = 5, and n=36n = 36: a36=12+(36−1)×5=12+35×5=12+175=187a_{36} = 12 + (36 – 1) \times 5 = 12 + 35 \times 5 = 12 + 175 = 187
Final answer:
187\boxed{187}
Explanation (300 words):
The sequence provided is 12, 17, 22, and so forth. By examining the difference between consecutive terms, we see a consistent increase of 5 each time: from 12 to 17 is +5, and from 17 to 22 is also +5. This consistent increment means the sequence is arithmetic.
An arithmetic sequence is defined by a starting value (called the first term, a1a_1) and a fixed step size (called the common difference, dd). Here, the first term a1a_1 is 12, and the common difference dd is 5.
To find any term in the sequence, the formula used is: an=a1+(n−1)da_n = a_1 + (n – 1)d
where ana_n is the nn-th term, a1a_1 is the first term, and dd is the common difference.
In this problem, we’re asked to find the 36th term. Substituting the known values, we calculate: a36=12+(36−1)×5=12+35×5=12+175=187a_{36} = 12 + (36 – 1) \times 5 = 12 + 35 \times 5 = 12 + 175 = 187
Thus, the 36th term in the sequence is 187.
Because this value is a whole number, there’s no need to round further. The answer is simply 187.
This approach can be used for any term in the sequence by changing the value of nn, making arithmetic sequences straightforward and predictable. Arithmetic sequences appear in many real-world situations where things increase or decrease by a constant amount regularly.