Predicting What Comes Next: Sequences and Progressions | Exercise 8.2

Question 2

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.

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Solution
Understand the Question
  • For an Arithmetic Progression (AP) with first term aa and common difference dd, the nn-th term is given by: tn=a+(n−1)dt_n = a + (n - 1)d
  • Given AP: 21,18,15,…21, 18, 15, \dots
    • First term, a=21a = 21
    • Common difference, d=18−21=−3d = 18 - 21 = -3
  • A number is a term of the AP if and only if solving tn=valuet_n = \text{value} yields a positive integer (natural number) for nn.

Step 1 · Find the general term of the AP

Given AP: 21,18,15,…21, 18, 15, \dots

First term, a=21a = 21

Common difference: d=18−21=−3d = 18 - 21 = -3

The nn-th term is: tn=a+(n−1)dt_n = a + (n - 1)d

Substitute a=21a = 21 and d=−3d = -3:

tn=21+(n−1)(−3)=21−3n+3=24−3n\begin{aligned} t_n &= 21 + (n - 1)(-3) \\ &= 21 - 3n + 3 \\ &= 24 - 3n \end{aligned}

Step 2 · Find the term number for −81-81

Set tn=−81t_n = -81:

24−3n=−81−3n=−81−24−3n=−105n=−105−3n=35\begin{aligned} 24 - 3n &= -81 \\ -3n &= -81 - 24 \\ -3n &= -105 \\ n &= \dfrac{-105}{-3} \\ n &= 35 \end{aligned}

Since n=35n = 35 is a positive integer, −81-81 is the 35th35^{\text{th}} term.

Step 3 · Check if 00 is a term of the AP

Set tn=0t_n = 0:

24−3n=03n=24n=243n=8\begin{aligned} 24 - 3n &= 0 \\ 3n &= 24 \\ n &= \dfrac{24}{3} \\ n &= 8 \end{aligned}

Since n=8n = 8 is a positive integer, 00 is the 8th8^{\text{th}} term of the AP.

Answer

−81-81 is the 35th35^{\text{th}} term, and 00 is the 8th8^{\text{th}} term of the AP.

Common Mistakes
  • Incorrect Sign of Common Difference: Taking d=21−18=3d = 21 - 18 = 3 instead of d=18−21=−3d = 18 - 21 = -3. Since the sequence is decreasing, dd must be negative.
  • Condition for Validity: For any value to be a term of an AP, the calculated value of nn must be a natural number (n∈Nn \in \mathbb{N}). If nn is a fraction or negative, that value is not a term.

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?

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