GSEB Class 7 Maths Solutions Chapter 12 બીજગણિતીય પદાવલિ Exercise 12.4

Question 1. Observe the number patterns made from identical line segments. Such divided numbers are found in electronics:
(a) 6, 11, 16, 21.... \( (5n+1) \)
(b) 4, 7, 10, 13.... \( (3n+1) \)
(c) 7, 12, 17, 22.... \( (5n+2) \)
If the number of digits created is taken as 1, the number of segments required to create 1 digit is given by an algebraic expression on the right side of each pattern. Like 6.40., how many segments will be needed to form 5, 10, and 100 digits?
Answer:
(a) For this pattern, the number of line segments is given by the expression \( 5n + 1 \).
When \( n = 5 \), the number of segments is \( 5(5) + 1 = 25 + 1 = 26 \).
When \( n = 10 \), the number of segments is \( 5(10) + 1 = 50 + 1 = 51 \).
When \( n = 100 \), the number of segments is \( 5(100) + 1 = 500 + 1 = 501 \).
(b) For this pattern, the number of line segments is given by the expression \( 3n + 1 \).
When \( n = 5 \), the number of segments is \( 3(5) + 1 = 15 + 1 = 16 \).
When \( n = 10 \), the number of segments is \( 3(10) + 1 = 30 + 1 = 31 \).
When \( n = 100 \), the number of segments is \( 3(100) + 1 = 300 + 1 = 301 \).
(c) For this pattern, the number of line segments is given by the expression \( 5n + 2 \).
When \( n = 5 \), the number of segments is \( 5(5) + 2 = 25 + 2 = 27 \).
When \( n = 10 \), the number of segments is \( 5(10) + 2 = 50 + 2 = 52 \).
When \( n = 100 \), the number of segments is \( 5(100) + 2 = 500 + 2 = 502 \).
In simple words: To find out how many segments are needed for any number of digits (n), you use the given algebraic rule for each pattern. Just put the number of digits (like 5, 10, or 100) into the 'n' part of the rule and then do the math.

Exam Tip: Carefully substitute the value of 'n' into the algebraic expression and perform the calculations step by step to avoid errors. Double-check your arithmetic, especially for larger numbers.

 

Question 2. Complete the table using the given algebraic expressions for 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 \( 7 \times 10 + 20 = 70 + 20 = 90 \).
The 100th term 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 \( (10)^2 + 1 = 100 + 1 = 101 \).
The 100th term is calculated as \( (100)^2 + 1 = 10000 + 1 = 10001 \).
In simple words: To complete the table, you need to use the algebraic rule given in each row. For each specific term (like the 5th, 10th, or 100th), replace 'n' in the expression with that term number. Then, perform the math operations to find the value of that term.

Exam Tip: Pay close attention to the exponent in \( n^2+1 \). Remember that squaring a number means multiplying it by itself, not by 2.