Pattern Analysis: A sequence of numbers is formed by adding 3, then 5, then 7, and so on. If the sequence starts with 2, find the 10th term in the sequence. Explain your reasoning.
Answer (2)
The 10th term in the sequence that starts with 2 and adds consecutive odd numbers is 101. This is found by systematically adding the odd numbers to the previous term. The final calculation to reach the 10th term includes several steps of addition.
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To find the 10th term in the sequence, let's carefully examine how the sequence is formed:
Start with the initial term : The sequence starts at 2.
Pattern of adding numbers : We add a sequence of odd numbers (3, 5, 7, etc.) to generate the next terms.
Identifying the sequence of additions :
1st step: Add 3 to the 1st term to get the 2nd term.
2nd step: Add 5 to the 2nd term to get the 3rd term.
3rd step: Add 7 to the 3rd term to get the 4th term.
So, each term is obtained by adding an increasing sequence of consecutive odd numbers to the previous term.
To find the 10th term in the sequence, we calculate the first few terms and identify the pattern:
1st term : a 1 = 2
2nd term : a 2 = a 1 + 3 = 2 + 3 = 5
3rd term : a 3 = a 2 + 5 = 5 + 5 = 10
4th term : a 4 = a 3 + 7 = 10 + 7 = 17
Continuing in this manner, let’s find the rest until the 10th term:
5th term : a 5 = a 4 + 9 = 17 + 9 = 26
6th term : a 6 = a 5 + 11 = 26 + 11 = 37
7th term : a 7 = a 6 + 13 = 37 + 13 = 50
8th term : a 8 = a 7 + 15 = 50 + 15 = 65
9th term : a 9 = a 8 + 17 = 65 + 17 = 82
10th term : a 10 = a 9 + 19 = 82 + 19 = 101
Thus, the 10th term of the sequence is 101. To summarize, by following the pattern of consecutively adding increasing odd numbers, we determine that the 10th term in the sequence is 101 .
