Ex 8.2, 5 - Chapter 8 - Predicting What Comes Next: Exploring Sequences & Progress

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Ex 8.2, 5 How many 2 -digit numbers are divisible by 3? What is the sum of all these 2-digit numbers? Numbers divisible by 3 are 3, 6, 9, 12, …. Let’s find Smallest and Largest 2-digit number divisible by 3 Smallest 2 digit number divisible by 3 Smallest two digit number divisible by 3 is 12 Largest 2 digit number divisible by 3 Since 99/3 = 33 So, 99 leaves remainder 0 when divided by 3 ∴ Largest 2 digit number divisible by 3 is 99 Now, our sequence starts with 12 and ends with 99 And, it will have a common difference of 3 (as all numbers are divisible by 3) So, our AP will be 12, 15, 18, 21, …., 99 We need to find How many 2 -digit numbers are divisible by 3 Thus, we need to find n Now, First term = a = 12 Common difference = d = 15 – 12 = 3 Last term = an = 99 Putting these values in formula an = a + (n – 1) d 99 = 12 + (n – 1) (3) 99 = 12 + 3n – 3 99 = 9 + 3n 3n = 99 – 9 3n = 90 n = 90/3 n = 30 ∴ There are 30 two-digit numbers divisible by 3 Now, we are asked What is the sum of all these 2-digit numbers So, we need to find Sn We know that Sn = 𝒏/𝟐[𝒂+𝒍] Putting n = 30, a = 12, l = 99 Sn = 𝟑𝟎/𝟐 [𝟏𝟐+𝟗𝟗] Sn = 15 × 111 Sn = 1665 Thus, sum of all these 2-digit numbers is 1665