Page 29 - Ganit Kaushal

Ex 3. Fill in the blanks.
1 2 3
(a) The next three missing terms of the sequence 1, ,, , ___ , ___ , ___
2 3 5

(b) The next three missing terms of the sequence 1, 5, 13, 25, ___ , ___ , ___
5 8 13 denominator
Sol. (a) , , . [ the pattern followed is .]
8 13 21 denominator + numerator

(b) 41, 61, 85. [ the pattern followed is – add 4, 8, 12, 16, 20, 24, …]
Ex 4. State whether the following statements are True (T) or False (F).

(a) If we add up and down the first hundred counting numbers we get 10,000.
(b) Snowflakes are made by snow.

Sol. (a) True. [ Ex 2, section 1.6]
(b) False [ They are fractal geometrical shapes made with the help of equilateral triangles]

Subjective/Competency Based Questions

Ex 5. Count the number of diagonals of the shape sequence of regular polygons. Do you see any pattern? If
yes, explain it. Based on this, find the number of diagonals in a heptagon, octagon and nonagon.
Sol. Number of diagonals in a regular/equilateral triangle = 0 [see fig. (i) below]

Number of diagonals in a regular quadrilateral = 2 [see fig. (ii) below]
Number of diagonals in a regular pentagon = 5 [5 different coloured lines in fig. (iii)]

Number of diagonals in a regular hexagon = 9 [3-red, 3-blue, 3-green lines in fig. (iv)]
So, we get 0, 2, 5, 9, and so on.

Diagonals: 0 2 5 9
(i) (ii) (iii) (iv)
Yes, we observe a pattern and the pattern is – add the counting numbers 2, 3, 4, 5, 6, 7, …

Based on this pattern, we conclude that:

The number of diagonals in a heptagon = 9 + 5 = 14.
The number of diagonals in an octagon = 14 + 6 = 20.
The number of diagonals in a nonagon = 20 + 7 = 27.

Ex 6. A thirsty monkey goes 10 steps down the stairs to drink water from a pond. After drinking the water,
the monkey happily comes back jumping upstairs taking at most two steps upwards at a time. In how
many different ways can the monkey come back to the top?

Sol. To climb 10 steps, the monkey has to make 9 jumps upwards to come to the top. So, the answer is the
9th term of the Virahānka sequence, i.e., 55.
\ The monkey can come back to the top in 55 different ways by taking at most 2 jumps upwards.

Patterns in Mathematics 27