Terms of sequences
15,22,31,42,55
find next 3 terms and the formula also for the sequence
the next three terms are 70,87,106. the pattern is adding consecutive odd numbers to each term.
ex: 15+7=22, 22+9=31, 31+11=42, 42+13=55 and so on.
So the formula for the nth term of the sequence is
10+4n+n^2
Well, let's see if I can keep up with those numbers and put on my clownerific thinking cap!
Looking at the sequence, I notice that each term seems to be increasing by a certain amount. So, let's try to figure out that pattern.
If we subtract the previous term from each term, we get the following differences:
22 - 15 = 7
31 - 22 = 9
42 - 31 = 11
55 - 42 = 13
Ah, it seems like the differences are increasing by 2 every time! That's interesting!
So, based on this pattern, the next differences would be:
13 + 2 = 15
15 + 2 = 17
17 + 2 = 19
Now, to find the next terms in the sequence, we can add these differences to the last term:
55 + 15 = 70
70 + 17 = 87
87 + 19 = 106
Hence, the next 3 terms in the sequence would be: 70, 87, 106.
As for the formula, let me put on my math clown wig and give it a go:
The terms seem to be increasing quadratically, so we can try a quadratic equation. Let's call the nth term in the sequence 'S_n'.
Based on our observations with the differences, we can say that the difference between each term and (n-1)th term is (2n - 1).
So, we can say that:
S_n = S_(n-1) + (2n - 1)
With S_1 being 15, we can start plugging in numbers and see if this formula holds!
S_2 = S_1 + (2*2 - 1) = 15 + 3 = 18, which is correct!
S_3 = S_2 + (2*3 - 1) = 18 + 5 = 23, which is correct!
Oh dear, sorry to say, but I think I'm just clowning around and struggling to find the right formula for this sequence. Maybe, there's a different pattern that's beyond my clownish comprehension.
To find the next three terms of the given sequence 15, 22, 31, 42, 55, let's first try to identify the pattern.
Looking at the numbers, we can observe that each term in the sequence is obtained by adding a specific number to the previous term.
To find this specific number, we can calculate the difference between consecutive terms:
22 - 15 = 7
31 - 22 = 9
42 - 31 = 11
55 - 42 = 13
We can see from this pattern that the differences between consecutive terms are increasing by 2 each time.
Therefore, the next difference is 13 + 2 = 15, and we can continue this pattern to find the next three terms:
55 + 15 = 70
70 + 17 = 87
87 + 19 = 106
So, the next three terms in the sequence are 70, 87, and 106.
In terms of the formula for this sequence, we can express it as follows:
nth term = a + (n - 1) * d
where:
- "a" is the first term of the sequence (15 in this case)
- "n" is the position of the term in the sequence
- "d" is the common difference between consecutive terms (2 in this case)
Using this formula, we can find any term in the sequence by plugging in the value of "n".