增长衰减公式中1的作用及等差数列求和除以2的原因咨询

Hey there! Let's unpack these two formula questions with straightforward, real-world examples—no confusing jargon required.

1. Why does the growth/decay formula A=P(1±r)^t include the number 1? #

Think of this formula as calculating the total amount after t periods, not just the change in amount. Let's use a concrete growth example:

• Suppose you have $200 (your principal P) with a 7% annual interest rate (r=0.07).

After one year, you don't just earn $14 in interest—you still have your original $200! Mathematically, that's:
$200 + ($200 * 0.07) = $200*(1 + 0.07)

The 1 here represents your original starting amount. If we left it out, we'd only calculate the interest earned ($200*0.07 = $14), which is just the growth portion—not the total money you have.

For decay, the logic is identical: if a car loses 12% of its value yearly, you're left with 88% of its original value. That's 1 - 0.12 = 0.88, so the total value becomes P*(1 - r) each year. The 1 ensures we're calculating the remaining total, not just the amount lost.

2. Why do we divide by 2 in the arithmetic series sum formula S(n) = \frac{n}{2}(a_1 + a_n)? #

This is easiest to see with the pairing method—let's use a simple sequence to demonstrate:

• Let's sum the first 6 integers: 1 + 2 + 3 + 4 + 5 + 6

Now write the same sum in reverse: 6 + 5 + 4 + 3 + 2 + 1. Add these two sums together:

(1+6) + (2+5) + (3+4) + (4+3) + (5+2) + (6+1) = 7 + 7 + 7 + 7 + 7 + 7 = 42

Each pair adds up to a_1 + a_n (the first term plus the last term), and there are exactly n pairs (6 pairs here). But wait—we just added the original sum twice! To get the actual sum of the original sequence, we divide that total by 2: 42 / 2 = 21, which is the correct sum.

For any arithmetic sequence, pairing terms from the start and end gives you n equal pairs. Since we doubled the sum to create these pairs, dividing by 2 gives us the true sum of the sequence.

内容的提问来源于stack exchange，提问作者Farhan Ali