Observe the following pattern and answer the following question:
1, 3, 6, 10, 15
a) Write the pattern rule for the given pattern.
The pattern is generated by consecutive integers.
b) Write the next three numbers in the pattern.
Answer (2)
The pattern represents triangular numbers generated by adding consecutive integers. The next three numbers in the pattern are 21, 28, and 36. This sequence can be found by continuously adding the next integer to the last number in the series.
;
To observe the pattern given: 1, 3, 6, 10, 15, we can notice that each number is a sum of consecutive natural numbers. This sequence is known as the sequence of triangular numbers.
a) Write the pattern rule for the given pattern.
The pattern can be described as:
First term (1): 1
Second term (3): 1 + 2 = 3
Third term (6): 1 + 2 + 3 = 6
Fourth term (10): 1 + 2 + 3 + 4 = 10
Fifth term (15): 1 + 2 + 3 + 4 + 5 = 15
The pattern rule is that each number in the sequence is formed by adding the next consecutive integer to the sum of all the previous integers. Mathematically, the n t h triangular number can be calculated using the formula:
T n = 2 n ( n + 1 )
b) Write the next three numbers in the pattern.
To find the next three numbers:
Sixth term: T 6 = 2 6 ( 6 + 1 ) = 2 6 × 7 = 21 So, the sixth number is 21.
Seventh term: T 7 = 2 7 ( 7 + 1 ) = 2 7 × 8 = 28 So, the seventh number is 28.
Eighth term: T 8 = 2 8 ( 8 + 1 ) = 2 8 × 9 = 36 So, the eighth number is 36.
Hence, the next three numbers in the pattern are 21, 28, and 36.
