NCERT Solutions for Class 9th Maths Chapter 8 Think and Reflect — Arithmetic Progressions
Book page 180 Updated on2026-09-08
Q1.
[Fig. 8.3, growing pattern of squares 1, 5, 9, 13] Can you predict the number of squares in Stages 5 and 6 of the sequence? In Stages 10, 11 and 12? In Stage 20? At any stage?
Answer
Yes — and once the rule is justified, every one of these follows by substitution. Each stage adds 4 squares, one to each arm of the X, so the counts are 1, 5, 9, 13, 17, 21, …
Going from Stage 4 to Stage 5 lengthens each of the four diagonal arms by one square: 13 + 4 = 17.
Why it happens: spotting "add 4" only lets you crawl forward one stage at a time. The formula comes from seeing how many times the 4 has been added: at Stage n the single central square has been joined by 4 squares on (n – 1) occasions, so the count is 1 + (n – 1) × 4. Expanding gives 4n – 3, and it is this step — from a pattern in the picture to an argument about the picture — that turns a guess into a rule you can trust at Stage 200.
Check it yourself: Stage 1 must give 1. Putting n = 1 in 4n – 3 gives 4 – 3 = 1. A formula that fails at n = 1 is the wrong formula.