Predicting What Comes Next: Sequences and Progressions | Exercise 8.2

Question 3

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

Check your answer with HomiSolve it yourself, then let Homi check your steps and spot mistakes.
Solution
Understand the Question
  • An Arithmetic Progression (AP) is a sequence where each term after the first is obtained by adding a constant value called the common difference (dd) to the preceding term.
  • For the sequence 11,8,5,2,…11, 8, 5, 2, \dots:
    • First term (aa or t1t_1) =11= 11
    • Common difference (dd) =t2−t1=8−11=−3= t_2 - t_1 = 8 - 11 = -3
  • The nthn^{\text{th}} term (explicit formula) is given by tn=a+(n−1)dt_n = a + (n-1)d.
  • The recursive rule expresses each term tnt_n using the previous term tn−1t_{n-1}, along with the base term t1t_1.

(i) Find the nthn^{\text{th}} term of the AP: 11,8,5,2,…11, 8, 5, 2, \dots.

Step 1 · Find the Common Difference and the nthn^{\text{th}} Term

For the given AP 11,8,5,2,…11, 8, 5, 2, \dots, the first term is a=11a = 11.Diagram 1

Calculate the common difference dd

d=8−11=−3\begin{aligned} d &= 8 - 11 \\[0.6em] &= -3 \end{aligned}

Using the formula for the nthn^{\text{th}} term tn=a+(n−1)dt_n = a + (n-1)d

tn=11+(n−1)(−3)=11−3n+3=14−3n\begin{aligned} t_n &= 11 + (n-1)(-3) \\[0.6em] &= 11 - 3n + 3 \\[0.6em] &= 14 - 3n \end{aligned}
Answer

(i) tn=14−3nt_n = 14 - 3n

(ii) Write the recursive rule for this AP.

Step 1 · Write the Recursive Rule

A recursive rule defines each term using the previous term, starting from the first term t1=11t_1 = 11.

tn=tn−1+d=tn−1−3\begin{aligned} t_n &= t_{n-1} + d \\[0.6em] &= t_{n-1} - 3 \end{aligned}

Therefore, the recursive rule is t1=11,tn=tn−1−3for n≥2t_1 = 11, \quad t_n = t_{n-1} - 3 \quad \text{for } n \ge 2

Answer

(ii) t1=11,tn=tn−1−3 for n≥2t_1 = 11, \quad t_n = t_{n-1} - 3 \text{ for } n \ge 2

Common Mistakes
  • Sign Error in Common Difference: Incorrectly calculating d=11−8=3d = 11 - 8 = 3 instead of d=t2−t1=8−11=−3d = t_2 - t_1 = 8 - 11 = -3. Since the AP decreases, dd must be negative.
  • Distributing Negative Numbers: Making a sign error when expanding (n−1)(−3)(n-1)(-3), such as writing −3n−3-3n - 3 instead of −3n+3-3n + 3.
  • Incomplete Recursive Rule: Forgetting to state the first term (t1=11t_1 = 11) or the domain (n≥2n \ge 2), both of which are required for a complete recursive definition.

More questions in Exercise 8.2

Q1

Find the 10th10^{\text{th}} and 26th26^{\text{th}} terms of the AP: 3,8,13,18,…3, 8, 13, 18, \dots

Q2

Which term of the AP: 21,18,15,…21, 18, 15, \dots is −81-81? Also, is 00 a term of this AP? Give reasons for your answer.

Q3

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

Q4

An AP consists of 50 terms in which the 3rd3^{\text{rd}} term is 12 and the last term is 106. Find the 29th29^{\text{th}} term.

(Hint: If aa is the first term and dd the common difference, then we arrive at the equations a+2d=12a + 2d = 12 and a+49d=106a + 49d = 106. Solve this pair of linear equations for aa and dd.)

Q5

How many 2-digit numbers are divisible by 3? What is the sum of all these 2-digit numbers?

Q6

Harish started work at an annual salary of ₹5,00,000 and received an increment of ₹20,000 each year. After how many years did his income reach ₹7,00,000?

Q7

A child arranges marbles in rows so that the first row has 1 marble, the second has 2 marbles, the third has 3, and so on up to 25 rows. How many marbles does the child use in all?

← Back to Predicting What Comes Next: Sequences and Progressions