Question 13 of 51intermediate🔧 ApplyNumerical4 marks

A sequence is given by the recursive rule t1=−5t_1 = -5, tn+1=tn+3t_{n+1} = t_n + 3 for n≥1n \geq 1. Find the first five terms of the sequence. Is 52 a term of this sequence? If so, which term is it?

Correct Answer

(First five terms) -5, -2, 1, 4, 7

(Is 52 a term? If so, which term?) Yes, it is the 20th term

Exercise: Exercise Set 8.1 | Q: 5 | (Page 6)
For More Understanding

Explanation

The recursive rule provides the first term and a method to find subsequent terms by adding a constant value (3) to the previous term. By calculating the first five terms, we establish the sequence. To determine if 52 is a term without calculating all terms up to 52, we convert the recursive rule into an explicit formula for the nth term, similar to the example in the context where the sequence 1, 4, 7... has the explicit rule t_n = 3n - 2. For this sequence starting at -5, the explicit rule is t_n = 3n - 8. Setting t_n = 52 allows us to solve for n, and since n is a positive integer, 52 is indeed a term of the sequence.

Solution Steps

  1. Step 1: Find the first five terms using the recursive rule. Given t1=−5t_1 = -5 and tn+1=tn+3t_{n+1} = t_n + 3:

    t1=−5t_1 = -5 t2=t1+3=−5+3=−2t_2 = t_1 + 3 = -5 + 3 = -2 t3=t2+3=−2+3=1t_3 = t_2 + 3 = -2 + 3 = 1 t4=t3+3=1+3=4t_4 = t_3 + 3 = 1 + 3 = 4 t5=t4+3=4+3=7t_5 = t_4 + 3 = 4 + 3 = 7

    The first five terms are -5, -2, 1, 4, 7.

  2. Step 2: Find the explicit rule for the nthn^{th} term. Since each term is 3 more than the previous term, the explicit formula is tn=t1+(n−1)×3t_n = t_1 + (n - 1) \times 3. tn=−5+3n−3=3n−8t_n = -5 + 3n - 3 = 3n - 8.

  3. Step 3: Check if 52 is a term of the sequence. Set tn=52t_n = 52:

    3n−8=523n - 8 = 52 3n=52+83n = 52 + 8 3n=603n = 60 n=20n = 20

    Since n=20n = 20 is a positive integer, 52 is a term of this sequence, and it is the 20th term.

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