Predicting What Comes Next: Sequences and Progressions | Exercise 8.2

Question 5

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

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Solution
Understand the Question
  • The two-digit numbers divisible by 33 form an Arithmetic Progression (AP): 12,15,18,…,9912, 15, 18, \dots, 99
  • In this AP:
    • First term (aa) =12= 12
    • Common difference (dd) =3= 3
    • Last term (ll or tnt_n) =99= 99
  • To find how many numbers there are, we use the nthn^{\text{th}} term formula: tn=a+(n−1)dt_n = a + (n - 1)d.
  • To find their sum, we use the sum formula: Sn=n2(a+l)S_n = \dfrac{n}{2}(a + l).

Step 1 · Find the Number of 2-Digit Numbers Divisible by 3

The list of 2-digit numbers divisible by 33 is 12,15,18,…,9912, 15, 18, \dots, 99.Diagram 1

This is an AP with:

  • First term, a=12a = 12
  • Common difference, d=3d = 3
  • Last term, tn=99t_n = 99

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

99=12+(n−1)399−12=3(n−1)87=3(n−1)873=n−129=n−1n=29+1n=30\begin{aligned} 99 &= 12 + (n - 1)3 \\ 99 - 12 &= 3(n - 1) \\ 87 &= 3(n - 1) \\ \dfrac{87}{3} &= n - 1 \\[0.6em] 29 &= n - 1 \\ n &= 29 + 1 \\ n &= 30 \end{aligned}

There are 3030 two-digit numbers divisible by 33.

Step 2 · Calculate the Sum of the Numbers

Using the sum formula for an AP Sn=n2(a+l)S_n = \dfrac{n}{2}(a + l)

Substitute n=30n = 30, a=12a = 12, and l=99l = 99

S30=302×(12+99)=15×111=1665\begin{aligned} S_{30} &= \dfrac{30}{2} \times (12 + 99) \\[0.6em] &= 15 \times 111 \\ &= 1665 \end{aligned}
Answer

Number of 2-digit numbers divisible by 3=303 = 30; Sum =1665= 1665

Common Mistakes
  • Wrong First/Last Term: Using 33 (not a 2-digit number) instead of 1212 as the first term, or 100100 instead of 9999 as the last term.
  • Off-by-One Error: Forgetting to add 11 when solving n−1=29n - 1 = 29, leading to n=29n = 29 instead of n=30n = 30.
  • Sum Formula Substitution: Using n=87n = 87 or forgetting to multiply by the factor n2\dfrac{n}{2}.

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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