Question

Find the sum of the first 20 terms of the AP $$1,3,5,7, \dots$$

Short answer

The sum of the first 20 terms of the arithmetic progression is 400.

Step-by-step solution

Identify the First Term

In an arithmetic progression (AP), the first term, denoted as 'a,' is the starting point of the sequence. For the given AP, the first term is 1.

Calculate the Common Difference

The common difference 'd' is the amount by which each term increases to get the next term. To find 'd', subtract the first term from the second term. For the given AP, the second term is 3, so the common difference is 3 - 1 = 2.

Sum of the First n Terms of an AP

The sum of the first n terms 'S_n' of an AP can be calculated using the formula: \(S_n = \frac{n}{2}(2a + (n - 1)d)\).

Substitute the Values and Solve

Substitute the values into the formula to find the sum of the first 20 terms: \(S_{20} = \frac{20}{2}(2 \times 1 + (20 - 1) \times 2)\).

Calculate the Sum

Carry out the calculations: \(S_{20} = 10(2 + 38) = 10(40) = 400\). The sum of the first 20 terms of the AP is 400.

Key concepts

Understanding the First Term of an AP

In any arithmetic progression (AP), the story begins with the very first term. Think of it as the story's protagonist from where everything else unfolds.
The first term, often represented by the variable 'a,' sets the initial value of the sequence. If we're talking about a line of dominoes, 'a' is that very first domino you tip over.

First Term in Action

When you meet the AP sequence of 1, 3, 5, 7, ..., that '1' isn't just a number; it's the cornerstone, our 'a'. It's from this first term that we start noticing the pattern that defines the progression. In this case, it seems the numbers are increasing by regular intervals, don't they? Imagine if we changed this first term, the entire sequence would transform. It's that pivotal.

Deciphering the Common Difference in an AP

Now let's chat about what keeps an arithmetic progression moving smoothly from one term to the next – the common difference, 'd'. It's like the steady beat in a song that dancers follow. You can find it by subtracting one term from the next one right after it.

Finding the Beat

To figure out the common difference in our sequence (1, 3, 5, 7, ...), grab the second term, which is 3, and subtract the first term, which is 1. Voilà, we have our 'd' which is 2. This means that to get from one term to the next, you just step up by 2. If the common difference were to change, the pace of our progression would change, too. It’s the secret sauce that gives an AP its unique flavor.

Calculating the Sum of an AP

When it's time to add up the first several terms of an arithmetic progression, it's like gathering fruits from a tree you've planted – you’re finding the sum of an AP. There is a nifty formula that helps you find this sum without having to add each term individually: \(S_n = \frac{n}{2}(2a + (n - 1)d)\).

Tallying Up the Harvest

Leveraging this formula, the sum of the first 'n' terms becomes a piece of cake. For our AP (1, 3, 5, 7, ...), we use our previously identified 'a' and 'd', plus the number of terms we’re interested in – let's say 20. Plugging these values into the formula, we get \(S_{20} = 10(40) = 400\). That’s the total of the first 20 terms. Without this formula, we'd be adding each term up one by one, and who has time for that?