What is the sum of the 22 terms of AP in which D is equal to 7 and 22nd term is 149?

Unlocking the Sum: An Arithmetic Progression Puzzle Solved

Have you ever encountered a math problem that seems a bit daunting at first glance? Perhaps you've seen something like this: "What is the sum of the 22 terms of an AP in which D is equal to 7 and the 22nd term is 149?" Don't worry, this isn't a test designed to stump you! Instead, it's an opportunity to understand a fundamental concept in mathematics: arithmetic progressions, often shortened to APs.

An arithmetic progression is simply a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is what we call the "common difference," and in our problem, it's represented by the letter D. In this specific case, we are told that D = 7.

We are also given that the 22nd term of this particular arithmetic progression is 149. Our goal is to find the sum of the first 22 terms of this sequence.

Breaking Down the Problem: What We Know

Let's list out the key pieces of information we have:

Number of terms (n): 22
Common difference (D): 7
The 22nd term (a_n or a₂₂): 149

The Formula for the nth Term of an AP

Before we can find the sum, it's helpful to recall (or learn!) the formula for finding any term in an arithmetic progression. This formula allows us to calculate any term if we know the first term and the common difference. It's expressed as:

a_n = a₁ + (n - 1) * D

Where:

a_n is the nth term
a₁ is the first term
n is the number of the term
D is the common difference

Finding the First Term (a₁)

We don't know the first term (a₁) directly, but we can use the information we have to find it. We know that the 22nd term (a₂₂) is 149, and we know n = 22 and D = 7. Let's plug these values into the nth term formula:

149 = a₁ + (22 - 1) * 7

Now, let's solve for a₁:

149 = a₁ + (21) * 7

149 = a₁ + 147

149 - 147 = a₁

2 = a₁

So, the first term of this arithmetic progression is 2.

The Formula for the Sum of an AP

Now that we know the first term, the last term (which is the 22nd term in this case), and the number of terms, we can calculate the sum of the 22 terms. There are a couple of common formulas for the sum of an arithmetic progression. One of them is particularly useful when you know the first term, the last term, and the number of terms:

S_n = n/2 * (a₁ + a_n)

Where:

S_n is the sum of the first n terms
n is the number of terms
a₁ is the first term
a_n is the nth term (the last term in our sequence)

Calculating the Sum

Let's plug in our known values into this sum formula:

n = 22
a₁ = 2
a₂₂ = 149

S₂₂ = 22/2 * (2 + 149)

Now, let's simplify:

S₂₂ = 11 * (151)

S₂₂ = 1661

Therefore, the sum of the first 22 terms of this arithmetic progression is 1661.

A Quick Recap of the Steps:

Identify the given values: number of terms (n), common difference (D), and the nth term (a_n).
Use the formula for the nth term to find the first term (a₁).
Use the formula for the sum of an arithmetic progression (S_n) to calculate the total sum.

It's fascinating how these formulas allow us to predict and calculate sums of sequences, even with a large number of terms, without having to list out every single number in between!

Frequently Asked Questions (FAQ)

How do I know if a sequence is an arithmetic progression?

A sequence is an arithmetic progression if the difference between any two consecutive terms is constant. You can check this by subtracting the first term from the second, the second from the third, and so on. If the difference is the same every time, it's an AP.

Why is the common difference important in an AP?

The common difference (D) is the backbone of an arithmetic progression. It dictates how each subsequent term is generated. Without a constant common difference, the sequence wouldn't be an arithmetic progression, and the formulas we use wouldn't apply.

Can I use a different formula to find the sum of an AP?

Yes! Another common formula for the sum of an AP is S_n = n/2 * [2*a₁ + (n - 1) * D]. This formula is useful if you don't know the last term but do know the first term, the number of terms, and the common difference. In our problem, we were given the last term, making the first formula we used more direct.