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NCERT Solutions for Class 9th Maths Chapter 8 Think and Reflect — Explicit Rule for a Sequence

Book page 177 Updated on2026-09-08

Q1.
Why is it useful to have an explicit formula for the nth term of a sequence?
Answer

Because it lets you reach any term directly, without first working out all the terms before it.

With un = 2n – 1:
u300 = 2 × 300 – 1 = 599 — in one step, not 299 steps

It is also the only convenient way to answer the reverse question. To ask "is 137 in this sequence, and where?" you simply solve the equation:

2n – 1 = 137
2n = 138
n = 69 — so 137 is the 69th odd number
Why it happens: an explicit rule is a function of the position. Because n appears on its own in the formula, substituting a value of n is a single calculation, and the formula can equally be read backwards as an equation in n. A recursive rule cannot do this: to reach the 300th term it must first know the 299th, which needs the 298th, and so on all the way down to the start.
Tip: when you solve tn = k, always look at the n you get. Only a natural number counts — a fraction or a negative value means k is not a term at all.
Q2.
Can you find the rule describing the nth term of the sequence of square numbers?
Answer

The sequence is 1, 4, 9, 16, 25, 36, … , and the rule is

tn = n2

Test it against the printed terms: t1 = 12 = 1, t2 = 22 = 4, t3 = 9, t4 = 16, t5 = 25, t6 = 36. All six agree.

Why it happens: the chapter has already shown that each square number is the sum of the odd numbers up to that position — 1 + 3 + 5 + … + (2n – 1). Fig. 8.2 shows why that sum is n2: the L-shaped bands assemble into an n × n array. So tn = n2 is not just a formula that fits the first six terms; it has a reason behind it, and that reason guarantees it for every n.
Check it yourself: the 20th square number should be 400, and 1 + 3 + 5 + … + 39 (twenty odd numbers) is indeed 400.
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