Penny doubling every day for 30 days demonstrates exponential growth, with the value rising sharply after day 25.

Let’s think through this step by step.

The problem is about a penny that doubles every day for 30 days. That means:

- Day 1: $0.01 (just one penny)
- Day 2: $0.02 (doubled from day 1)
- Day 3: $0.04 (doubled from day 2)
- Day 4: $0.08
- ...
- And so on, until Day 30.

This is exponential growth — each day you multiply the previous day’s amount by 2.

We can write this as a formula:

Amount on Day n = 0.01 × 2^(n−1)

Why? Because on Day 1, we haven’t doubled yet — it’s just the starting penny. So we use exponent (n−1).

Let’s test it:

- Day 1: 0.01 × 2^0 = 0.01 × 1 = $0.01 ✔
- Day 2: 0.01 × 2^1 = 0.01 × 2 = $0.02 ✔
- Day 3: 0.01 × 2^2 = 0.01 × 4 = $0.04 ✔

Good. Now let’s find the amount on Day 30:

→ Amount = 0.01 × 2^(30−1) = 0.01 × 2^29

Now calculate 2^29.

We know:

2^10 = 1,024
2^20 = (2^10)^2 = 1,024 × 1,024 = 1,048,576
2^29 = 2^20 × 2^9 = 1,048,576 × 512

Let’s compute that:

1,048,576 × 512

Break it down:

= 1,048,576 × (500 + 12)
= (1,048,576 × 500) + (1,048,576 × 12)

First part:
1,048,576 × 500 = 1,048,576 × 5 × 100 = 5,242,880 × 100 = 524,288,000

Second part:
1,048,576 × 12 = 1,048,576 × 10 + 1,048,576 × 2 = 10,485,760 + 2,097,152 = 12,582,912

Add them:

524,288,000 + 12,582,912 = 536,870,912

So 2^29 = 536,870,912

Now multiply by 0.01:

→ 536,870,912 × 0.01 = 5,368,709.12

That’s in dollars.

Wait — but look at the graph. The y-axis goes up to over 5 million, and on day 30 it’s near 5.3 million — which matches!

But hold on — sometimes people get confused whether it’s “doubling every day” meaning including day 1 or not. Let’s double-check with small numbers.

If you start with 1 cent on day 1, then:

Day 1: 1¢
Day 2: 2¢
Day 3: 4¢
...
Day 30: 2^29 cents

Yes — because day 1 is 2^0 cents, day 2 is 2^1, ..., day 30 is 2^29 cents.

So total cents on day 30 = 2^29 = 536,870,912 cents

Convert to dollars: divide by 100 → $5,368,709.12

✔ This matches our earlier calculation.

Also, note: the graph shows the value rising sharply after day 25 — which makes sense because exponential growth starts slow and explodes later.

Final check: Is there any trick here? Sometimes problems say “you get a penny today, and it doubles tomorrow”, but here it says “a penny doubling every day for 30 days” — and the graph starts at day 1 with almost zero (which is 1 cent), so our interpretation is correct.

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Final Answer:
$5,368,709.12