Write the next term of sequence,and then write the rule for the nth term.
1)1,8,27,64,....125
the nth is
2) 3,8,15, 24,.....35
the nth is
Can you help me and correct ? thank
1)
a (n ) = n ^ 3
1 ^ 3 = 1
2 ^ 3 = 8
3 ^ 3 = 27
4 ^ 3 = 64
5 ^ 3 = 125
6 ^ 3 = 216
2 )
a (n ) = n ^ 2 + 2 n
1 ^ 2 + 2 * 1 = 1 + 2 = 3
2 ^ 2 + 2 * 2 = 4 + 4 = 8
3 ^ 2 + 2 * 3 = 9 + 6 = 15
4 ^ 2 + 2 * 4 = 16 + 8 = 24
5 ^ 2 + 2 * 5 = 25 + 10 = 35
6 ^ 2 + 2 * 6 = 36 + 12 = 48
Sure, I can help you with that!
1) The given sequence is 1, 8, 27, 64, and the next term is 125. To find the rule for the nth term, we can observe that each term is obtained by cubing the corresponding natural number. So, the pattern is that each term is the cube of n, where n represents the position of the term in the sequence. Therefore, the rule for the nth term of this sequence is n^3.
2) The given sequence is 3, 8, 15, 24, and the next term is 35. To find the rule for the nth term, we can observe that there is an increment in the differences between the terms. The differences between consecutive terms are 5, 7, 9. These differences are increasing by 2 each time. So, this indicates that the sequence is increasing by adding consecutive odd numbers (starting from 5) to the previous term.
Let's break it down further:
3 + 5 = 8
8 + 7 = 15
15 + 9 = 24
24 + 11 = 35
From this pattern, we can conclude that the common difference between consecutive terms increases by 2 each time, starting from 5. Therefore, the rule for the nth term of this sequence is given by n^2 + (n + 1).
I hope this helps clarify the pattern and rules for the given sequences!