Observe the pattern below. How many squares will be there in Step 4, Step 10, Step 50? Class 7
hardObserve the pattern below. How many squares will be there in Step 4, Step 10, Step 50? Class 7
Question 1.
Observe the pattern below. How many squares will be there in Step 4, Step 10, Step 50? Write a general formula. How would the formula change if we want to count the number of vertices of all the squares. Class 7
Solution:
Pattern is as follows : 5, 9, 13, ...
At step 4, number of squares = 13 + 3 = 17.
At step 10, number of squares = 5 + 9 × 4 = 41.
At step 50, number of squares = 5 + 49 × 4 = 5 + 196 = 201.
General formula, at step n, number of squares = 5 + (n - 1) × 4 = 5 + 4n - 4 = 4n + 1.
For vertices, the pattern is as follows :
16, 16 + 4 × 3, 16 + 4 × 3 + 4 × 3, ...
i. e., 16, 16 + 12, 16 + 24, ...
i. e., 16 + (n - 1) × 12 = 16 + 12n - 12
= 12n + 4.
Question 2.
Numbers are written in a particular sequence in this endless 4-column grid.
Do you see any pattern in it? List other patterns that you see.
(a) Give expressions to generate all the numbers in a given column (1, 2, 3, 4).
Solution:
In column 1, pattern is 1, 5, 9,...
In column 2, pattern is 2, 6, 10, ...
In column n, pattern will be n, n + 4, n + 8,...
The expression can be n, n + 4, n + 2 × 4, n + 3 v 4, ... n + m × 4.
i.e., n, n + 4, n + 2 × 4, ... n + 4m.
(b) In which row and column will the following numbers appear :
(i) 124 (ii) 147 (iii) 201
Solution:
(i) 124 will appear in 4th column and 31st row, because 124 = 4 + 30 × 4.
(ii) 147 will appear in 3rd column and 36th row, because 147 = 3 + 36 × 4.
(iii) 201 will appear in 1st column and 51th row, because 201 = 1 + 50 × 4.
(c) What number appears in row r and column c ?
Solution:
In row r and column c, number appear will be c + 4 (r - 1).
(d) Observe the positions of multiples of 3.
Solution:
These patterns are as follows :
(i) 3, 6, 9
(ii) 12, 15, 18, 21
(iii) 24, 27, 30, 33
(iv) 36, 39, 42, 45
(v) 48, 51, 54, 57 and so on. (See figure)
Other patterns:
(i) From top right towards bottom left, patterns are:
1, 6, 11, 16;
5, 10, 15, 20;
9, 14, 19, 24; and so on. Difference of 5
(ii) From top left to bottom right, patterns are :
2, 5:
3, 6, 9;
4, 7, 10, 13;
8, 11, 14, 17;
12, 15, 18, 21; and so on. Difference of 3.