Question

The first four patterns of a sequence of figures made of toothpicks are as shown:
K
KKKH
수
Pattern 1
Pattern 2
Pattern 3
Pattern 4
On the basis of the above information, answer the following questions :
(i) How many toothpicks are there in the 15th patterns?
(a) 46
(c) 42
(ii) How many toothpicks are there in the 20th patterns?
(a) 650
(b) 44
(d) 38
(b) 310​

Answer 1

Solution :-

→ First pattern of figure is made up of = 4 toothpicks .

→ Second pattern of figure is made up of = 7 toothpicks .

→ Third pattern of figure is made up of = 10 toothpicks .

→ Fourth pattern of figure is made up of = 13 toothpicks .

so, we can conclude that, 4,7,10,13 ______ form an AP series .

• first term = a = 4
• common difference = d = 7 - 4 = 3 .

then,

→ T(n) = a + (n - 1)d

→ T(15) = 4 + (15 - 1)3

→ T(15) = 4 + 14 * 3

→ T(15) = 4 + 42

→ T(15) = 46 (Ans.a)

Therefore, there are 46 toothpicks in the 15th patterns .

and,

→ T(n) = a + (n - 1)d

→ T(20) = 4 + (20 - 1)3

→ T(20) = 4 + 19 * 3

→ T(20) = 4 + 57

→ T(20) = 61 (Ans.c)

Therefore, there are 61 toothpicks in the 20th patterns .

also,

→ T(n) = a + (n - 1)d

→ 136 = 4 + (n - 1)3

→ 136 = 4 + 3n - 3

→ 136 = 3n + 1

→ 3n = 136 - 1

→ 3n = 135

→ n = 45 (Ans.d)

Therefore, the value of n is 45 .