what is a rule of next three terms of 6, 10, 15, 21, ....., ......, .....
b(k) means k th term
b(k+1)=b(k)+3+k
written as a recursive :
t(1) = 6
t(n) = t(n-1) + k+1, where k >1
as a function of n,
double the terms: 12, 20, 30, 42 ...
12 = 3*4
20 = 4*5
30 = 5*6
42 = 6*7
looks like
t(n) = (n+2)(n+3)/2 <--- remember we doubled
To determine the rule for the next three terms in the sequence 6, 10, 15, 21, ... , ... , ... , we need to look for a pattern or a relationship between the numbers.
Looking at the given sequence, we can observe that each term is obtained by adding consecutive increasing positive integers. Let's break it down:
The first term is 6, which is obtained by adding the integers 1 and 5 (1 + 5 = 6).
The second term is 10, which is obtained by adding the integers 1, 2, and 7 (1 + 2 + 7 = 10).
The third term is 15, which is obtained by adding the integers 1, 2, 3, and 9 (1 + 2 + 3 + 9 = 15).
The fourth term is 21, which is obtained by adding the integers 1, 2, 3, 4, and 11 (1 + 2 + 3 + 4 + 11 = 21).
Based on this pattern, we can see that each term is obtained by adding the integers consecutively, starting from 1 and increasing by 1 for each subsequent term. The last term is obtained by adding the integer 1, 2, 3, 4, ... and so on.
Therefore, the next three terms in the sequence would be:
... , ... , ...
To find the next term, we continue the pattern described above. Adding 1, 2, 3, and 5, we get:
21 + 1 + 2 + 3 + 5 = 32
So, the next term would be 32.
Continuing the pattern, we find the following two terms:
32 + 1 + 2 + 3 + 4 + 7 = 49
49 + 1 + 2 + 3 + 4 + 5 + 9 = 73
Therefore, the next three terms in the sequence are 32, 49, and 73.