Look for a pattern and find the next two numbers in the sequence and then write a general version of the nth term.
1 2 3 4 5 6 7 8 n
18 30 42 54 66 78
I can recognize the pattern as being + 12 so under 7 my answer would be 90 and under 8 my answer would be 102, but how do I figure out what the nth term would be?
As n increases by 1, the terms increases by 12. That means that if you add n to the term number, you add 12n to the value. So, start with
Tn = 12n
But T1=18, not 12. So, you need to add 6:
Tn = 12n+6
Or, you can use the general formula for an arithmetic progression with first term=a and common difference=d:
Tn = a + (n-1)d
You know that d=12 and a=18, so
Tn = 18 + (n-1)*12
= 18 + 12n - 12
= 6+12n
To find the nth term of the given sequence, we first need to identify the pattern. In this case, we can observe that each term in the sequence is obtained by multiplying the position of the term by 12.
Let's break it down step by step:
1. Write down the sequence:
1, 2, 3, 4, 5, 6, 7, 8
2. Write down the corresponding terms in the second row:
18, 30, 42, 54, 66, 78
3. Identify the difference between the terms in the second row:
The differences are 12, 12, 12, 12, 12
4. Write down the differences in the third row:
12, 12, 12, 12, 12
5. Now, the third row gives us a constant difference of 12 between each term. This tells us that the sequence follows an arithmetic progression.
6. To find the next term after 8, we can simply add 12:
78 + 12 = 90
7. Similarly, to find the term after 90, we add 12 again:
90 + 12 = 102
From this pattern, we can generalize the nth term using the formula for arithmetic progressions:
nth term = a + (n - 1) * d
where:
- a is the first term in the sequence (18 in this case)
- n is the position of the term in the sequence
- d is the common difference between consecutive terms (12 in this case)
So, the general formula for the nth term in this sequence is:
nth term = 18 + (n - 1) * 12
By substituting the value of n, you can find any term in the sequence.