Ramanan opens a store. On the first day he sold 12 juices. On the second day he sold 12 more than the first day. This continues. Every day he sold 12 more juices than the first day. On which day will he sell juice to his 300th customer 

To solve this problem, let's first define the situation mathematically.

- Let \( n \) represent the day number.
- On the first day (\( n = 1 \)), Ramanan sold 12 juices.
- On the second day (\( n = 2 \)), he sold \( 12 + 12 = 24 \) juices.
- On the third day (\( n = 3 \)), he sold \( 12 + 24 = 36 \) juices.

The pattern here suggests that the number of juices sold on day \( n \) is \( 12n \).

Next, we need to determine on which day the 300th juice will be sold. The total number of juices sold up to day \( n \) can be expressed as the sum of an arithmetic series:

\[
S_n = 12 \times 1 + 12 \times 2 + 12 \times 3 + \ldots + 12 \times n
\]

This is equivalent to:

\[
S_n = 12(1 + 2 + 3 + \ldots + n)
\]

The sum \( 1 + 2 + 3 + \ldots + n \) is the sum of the first \( n \) natural numbers, which is given by:

\[
\text{Sum} = \frac{n(n+1)}{2}
\]

Thus, the total number of juices sold by day \( n \) is:

\[
S_n = 12 \times \frac{n(n+1)}{2} = 6n(n+1)
\]

We want to find the smallest \( n \) such that \( S_n \geq 300 \):

\[
6n(n+1) \geq 300
\]

Let's solve this inequality:

\[
n(n+1) \geq \frac{300}{6} = 50
\]

Now, we need to find the smallest \( n \) such that \( n(n+1) \geq 50 \).

This inequality can be solved by testing values of \( n \):

\[
\text{For } n = 6, \quad 6 \times 7 = 42 \quad (\text{which is less than 50})
\]
\[
\text{For } n = 7, \quad 7 \times 8 = 56 \quad (\text{which is greater than or equal to 50})
\]

So, the smallest \( n \) that satisfies the inequality is \( n = 7 \).

Therefore, **on the 7th day, Ramanan will sell juice to his 300th customer**.

Would you like further details on this, or do you have any other questions? Here are five related questions you might find interesting:

1. How many juices will Ramanan have sold by the end of the 7th day?
2. What is the total number of juices sold on the 6th day?
3. On which day will Ramanan sell his 500th juice?
4. If Ramanan sells 15 more juices each day instead of 12, on which day will he reach 300 juices?
5. What if Ramanan sold only 10 juices on the first day and increased by 10 juices each day? How would that change the result?

**Tip:** When dealing with arithmetic sequences, knowing how to sum the sequence quickly can save time in solving related problems.

Ramanan opens a store. On the first day he sold 12 juices. On the second day he sold 12 more than the first day. This continues. Every day he sold 12 more juices than the first day. On which day will he sell juice to his 300th customer?

Discover how to solve a problem involving arithmetic sequences where Ramanan sells increasing numbers of juices each day. Learn the step-by-step solution to determine on which day he will serve his 300th customer. This problem covers core algebraic concepts and is suitable for students in Grades 6-8.

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