Page 23 - Ganit Kaushal

Ex 6. Observe the following sequence/pattern of letter Es formed by matchsticks and answer the following
questions:

......
(i) (ii) (iii)
(a) How many matchsticks are needed in the 5th figure of this sequence?

(b) How many letters E will be formed with the help of 120 matchsticks?
(c) Give the formula that represents the n-th general term/figure of this sequence/pattern.
Sol. Observe that it takes five matchsticks to make one E. To answer all the above questions, let us first
make the following table that tells us about the number of Es formed (i.e., the number of terms) and
the matchsticks needed (i.e., the value of that term).

Number of Es formed 1 2 3 4 5 6 ……
Number of matchsticks required 5 10 15 20 25 30 ……

Note that–
5 = 5 × 1, 10 = 5 × 2, 15 = 5 × 3, 20 = 5 × 4, 25 = 5 × 5, 30 = 5 × 6, and so on.

We can now generalise this information to write the nth term T of this sequence as; T = 5n, where ‘n’ is the
n
n
number of Es formed and T represents the number of matchsticks required to make ‘n’ Es. This is the answer
n
to (c) above.
Now, we can easily tell the 5th term of the sequence. It contains 5 × 5 = 25 (put n = 5 in T ) matchsticks. This
n
answers (a).

Again, 120 = 5 × 24, therefore, 24 Es will be formed with the help of 120 matchsticks. This answers (b).
Ex 7. Observe the following pattern of squares formed by matchsticks.

(i) (ii) (iii)
Based on this pattern, answer the following questions:
(a) How many matchsticks are needed in the 8th figure (term) of this sequence?

(b) How many squares will be formed with the help of 61 matchsticks?
(c) Give the formula that represents the n-th general term of this pattern/sequence.
Sol. To answer all the above questions, let us first make the following table:

Number of squares formed 1 2 3 4 5 6 ……

Number of matchsticks required 4 7 10 13 16 19 ……
Note that–

4 = 3 × 1 + 1, 7 = 3 × 2 + 1, 10 = 3 × 3 + 1, 13 = 3 × 4 + 1, 16 = 3 × 5 + 1, and so on.
We can now generalise this information as: T = 3n + 1, where ‘n’ is the number of squares formed and T
n
n
represents the number of matchsticks required to make ‘n’ squares. This is the answer to (c) above.

Patterns in Mathematics 21