Question 24 of 51intermediate🔧 ApplyShort Answer3 marks

Find the nthn^{th} term of the AP: 11, 8, 5, 2 … Write the recursive rule for this AP.

Correct Answer

(nth_term) Given AP is 11, 8, 5, 2, ... The first term (aa) is 11 and the common difference (dd) is 8−11=−38 - 11 = -3. To find the nthn^{th} term, we use the formula tn=a+(n−1)dt_n = a + (n-1)d. Substituting the values: tn=11+(n−1)(−3)=11−3n+3=14−3nt_n = 11 + (n-1)(-3) = 11 - 3n + 3 = 14 - 3n.

(recursive_rule) The recursive rule for an AP relates each term to its previous term. The general recursive rule is t1=at_1 = a, tn=tn−1+dt_n = t_{n-1} + d for n≥2n \geq 2. For this AP, since a=11a = 11 and d=−3d = -3, the recursive rule is t1=11t_1 = 11, tn=tn−1−3t_n = t_{n-1} - 3 for n≥2n \geq 2.

Exercise: Exercise Set 8.2 | Q: 3 | (Page 12)
For More Understanding

Explanation

The student identifies the first term and common difference from the given AP. Using the formula for the nth term of an AP from the context, they calculate the explicit formula. Then, using the recursive rule structure provided in the text, they substitute the specific values for this AP.

Solution Steps

  1. Step 1: Identify first term (a) = 11 and common difference (d) = 8 - 11 = -3.

  2. Step 2: Apply the nth term formula t_n = a + (n-1)d. So, t_n = 11 + (n-1)(-3) = 14 - 3n.

  3. Step 3: Apply the general recursive rule t_1 = a, t_n = t_{n-1} + d for n >= 2. Substituting a and d gives t_1 = 11, t_n = t_{n-1} - 3 for n >= 2.

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