Question 11 of 51intermediate🔧 ApplyShort Answer3 marks

Determine whether 97 and 172 are terms of the sequence tn=5n−3t_n = 5n - 3 for n≥1n \geq 1.

Correct Answer

(97) To determine whether 97 is a term of the sequence tn=5n−3t_n = 5n - 3, we equate tnt_n to 97 and solve for nn. 5n - 3 = 97 leads to 5n = 100 and n = 20. Since n=20n = 20 is a natural number, 97 is the 20th term of the sequence.

(172) Similarly, to check for 172, we equate tnt_n to 172 and solve for nn. 5n - 3 = 172 leads to 5n = 175 and n = 35. Since n=35n = 35 is also a natural number, 172 is the 35th term of the sequence.

Exercise: Exercise Set 8.1 | Q: 3 | (Page 6)
For More Understanding

Explanation

The method to check if a number belongs to a sequence is to set the general term equal to the number and solve for n. If n is a natural number (n ≥\geq 1), the number is a term. The textbook context illustrates this: solving 5n - 2 = 308 gives n = 62 (a natural number, so it's a term), while 5n - 2 = 471 gives n = 94.6 (not a natural number, so not a term). Applying this to 97 and 172 yields n = 20 and n = 35 respectively, both natural numbers.

Solution Steps

  1. Step 1: For 97, set 5n - 3 = 97. Solving gives 5n = 100, so n = 20. Since 20 is a natural number, 97 is the 20th term.

  2. Step 2: For 172, set 5n - 3 = 172. Solving gives 5n = 175, so n = 35. Since 35 is a natural number, 172 is the 35th term.

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