- every 6th customer at a flower shop receives a free rose and every 9th customer relieves a free lily.
- If first term of a G.P. is 20 and common ratio is 4. Find the 5th term.
a) 10240
b) 40960
c) 5120
d) 2560.
The Correct Answer and Explanation is :
Problem 1: Flower Shop Customer Rewards
In the first scenario, we need to determine the pattern of customer rewards at a flower shop. Every 6th customer receives a free rose, and every 9th customer receives a free lily.
To understand the pattern, we can think of it as a system of multiples:
- Every 6th customer: Customers numbered 6, 12, 18, 24, etc. will receive a free rose.
- Every 9th customer: Customers numbered 9, 18, 27, 36, etc. will receive a free lily.
To determine if a customer receives both a rose and a lily, we need to find common multiples of 6 and 9. The Least Common Multiple (LCM) of 6 and 9 is 18. Therefore, every 18th customer will receive both a rose and a lily. These customers are numbered 18, 36, 54, etc.
So, in summary:
- Every 6th customer gets a rose.
- Every 9th customer gets a lily.
- Every 18th customer receives both a rose and a lily.
Problem 2: Finding the 5th Term of a Geometric Progression (G.P.)
Given the first term of the geometric progression (G.P.) is 20 and the common ratio is 4, we can calculate the 5th term using the formula for the nth term of a G.P.: Tn=a⋅r(n−1)T_n = a \cdot r^{(n-1)}
Where:
- aa is the first term (20),
- rr is the common ratio (4),
- nn is the term number (5).
Substituting the known values: T5=20⋅4(5−1)=20⋅44=20⋅256=5120T_5 = 20 \cdot 4^{(5-1)} = 20 \cdot 4^4 = 20 \cdot 256 = 5120
Therefore, the 5th term is 5120.
Correct answer: c) 5120
Explanation of the G.P. Calculation
In a geometric progression, each term after the first is found by multiplying the previous term by a constant, known as the common ratio. In this case, the first term is 20, and each subsequent term is obtained by multiplying the previous term by 4.
To find the 5th term:
- Start with 20 (the first term).
- Multiply by 4 to get the second term: 20×4=8020 \times 4 = 80.
- Multiply by 4 again to get the third term: 80×4=32080 \times 4 = 320.
- Multiply by 4 to get the fourth term: 320×4=1280320 \times 4 = 1280.
- Finally, multiply by 4 to get the fifth term: 1280×4=51201280 \times 4 = 5120.
Thus, the fifth term in the progression is 5120.