Question 7 of 51intermediate🔍 AnalyzeShort Answer2 marks

Can you find the rule describing the nthn^{th} term of the sequence of square numbers?

Correct Answer

The sequence of square numbers begins as 1, 4, 9, 16, 25, and so on. By observing the terms carefully, we can identify a clear pattern relating each term to its position in the sequence.

The 1st term is 12=11^2 = 1, the 2nd term is 22=42^2 = 4, the 3rd term is 32=93^2 = 9, and the 4th term is 42=164^2 = 16. Following this consistent pattern, the nthn^{th} term is the square of its position number, nn. Thus, the explicit rule describing the nthn^{th} term of the sequence of square numbers is given by the formula tn=n2t_n = n^2.

Exercise: Think and Reflect | Q: 1 | (Page 4)
For More Understanding

Explanation

The question asks for the rule of the sequence of square numbers. Although the context discusses arithmetic progressions and explicit rules like tn=3n−2t_n = 3n - 2 or sn=5n−2s_n = 5n - 2, the 'Think and Reflect' prompt specifically asks the student to find the rule for square numbers. A square number is the product of an integer with itself. By listing the first few square numbers (1, 4, 9, 16...) and observing their positions (1st, 2nd, 3rd, 4th...), the pattern emerges where each term equals the square of its position index. Therefore, the explicit formula for the nth term is tn=n2t_n = n^2.

Solution Steps

  1. Step 1: Identify the sequence of square numbers as 1, 4, 9, 16, 25, ...

  2. Step 2: Observe the relationship between the term and its position: 1st term = 121^2, 2nd term = 222^2, 3rd term = 323^2, 4th term = 424^2.

  3. Step 3: Generalize the pattern for the nthn^{th} position, which gives the explicit rule tn=n2t_n = n^2.

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