GSEB Class 7 Maths Solutions Chapter 12 Algebraic Expressions Exercise 12.4

Question 1. Observe the patterns of digits made from line segments of equal length. You will find such segmented digits on the display of electronic watches or calculators.
(a) For the digit 6, the pattern is 6, 11, 16, 21... which follows the rule \( (5n + 1) \).
(b) For the digit 4, the pattern is 4, 7, 10, 13... which follows the rule \( (3n + 1) \).
(c) For the digit 8, the pattern is 7, 12, 17, 22... which follows the rule \( (5n + 2) \).
If the number of digits formed is taken to be n, the number of segments required to form n digits is given by the algebraic expression appearing on the right of each pattern. How many segments are required to form 5, 10, 100 digits of the kind 6, 4, 8.

Answer:
(a) For the digit 6:
The expression for the number of line segments needed to form 'n' figures of the digit 6 is \( 5n + 1 \).
For 5 figures, the number of line segments will be \( 5 \times 5 + 1 = 25 + 1 = 26 \).
For 10 figures, the number of line segments will be \( 5 \times 10 + 1 = 50 + 1 = 51 \).
For 100 figures, the number of line segments will be \( 5 \times 100 + 1 = 500 + 1 = 501 \).
(b) For the digit 4:
The expression for the number of line segments needed to form 'n' figures of the digit 4 is \( 3n + 1 \).
For 5 figures, the number of line segments will be \( 3 \times 5 + 1 = 15 + 1 = 16 \).
For 10 figures, the number of line segments will be \( 3 \times 10 + 1 = 30 + 1 = 31 \).
For 100 figures, the number of line segments will be \( 3 \times 100 + 1 = 300 + 1 = 301 \).
(c) For the digit 8:
The expression for the number of line segments needed to form 'n' figures of the digit 8 is \( 5n + 2 \).
For 5 figures, the number of line segments will be \( 5 \times 5 + 2 = 25 + 2 = 27 \).
For 10 figures, the number of line segments will be \( 5 \times 10 + 2 = 50 + 2 = 52 \).
For 100 figures, the number of line segments will be \( 5 \times 100 + 2 = 500 + 2 = 502 \).
In simple words: To find out how many segments are needed for any number of digits, just put the number of digits into the right formula for each digit type (6, 4, or 8) and calculate.

Exam Tip: Remember to use the correct algebraic expression for each digit type to avoid calculation errors. Always double-check your arithmetic.

 

Question 2. Use the given algebraic expression to complete the table of number patterns.

Answer:
(i) For the expression \( 2n - 1 \):
The 100th term is calculated as \( 2 \times 100 - 1 = 200 - 1 = 199 \).
(ii) For the expression \( 3n + 2 \):
The 5th term is calculated as \( 3 \times 5 + 2 = 15 + 2 = 17 \).
The 10th term is calculated as \( 3 \times 10 + 2 = 30 + 2 = 32 \).
The 100th term is calculated as \( 3 \times 100 + 2 = 300 + 2 = 302 \).
(iii) For the expression \( 4n + 1 \):
The 5th term is calculated as \( 4 \times 5 + 1 = 20 + 1 = 21 \).
The 10th term is calculated as \( 4 \times 10 + 1 = 40 + 1 = 41 \).
The 100th term is calculated as \( 4 \times 100 + 1 = 400 + 1 = 401 \).
(iv) For the expression \( 7n + 20 \):
The 5th term is calculated as \( 7 \times 5 + 20 = 35 + 20 = 55 \).
The 10th term is calculated as \( 7 \times 10 + 20 = 70 + 20 = 90 \).
For the 100th term, it is calculated as \( 7 \times 100 + 20 = 700 + 20 = 720 \).
(v) For the expression \( n^2 + 1 \):
The 5th term is calculated as \( 5^2 + 1 = 25 + 1 = 26 \).
The 10th term is calculated as \( 10^2 + 1 = 100 + 1 = 101 \).
In simple words: To fill in the table, replace 'n' in each expression with the term number (like 5 for the 5th term, 10 for the 10th term, or 100 for the 100th term) and then do the math.

Exam Tip: Pay close attention to the order of operations (PEMDAS/BODMAS) when evaluating expressions, especially with exponents like \( n^2 \).

Note:

1. We can also create some patterns using dots, such as:

\( 1^2 = 1 \)

\( 2^2 = 4 \)

\( 3^2 = 9 \)

\( 4^2 = 16 \)

2. We can also have some geometrical patterns as: