Function Evaluation And Arithmetic Sequences Problem Solving Guide

Hey guys! Let's dive into some cool math problems and break them down together. We've got a function evaluation and an arithmetic sequence question to tackle. So, grab your thinking caps, and let’s get started!

Q3. Evaluating Functions Made Easy: A Step-by-Step Guide

Let's kick things off with our function question. Function evaluation is a fundamental concept in algebra, and mastering it is super important for more advanced math topics. Our function is defined as g(x) = 2x³ + 2x² - 7x + 5, and the mission is to find the value of g(0). Sounds like a plan, right?

Understanding the Function

First, let's really get what this function is all about. A function, in simple terms, is like a machine. You pop in a number (x in this case), and the machine spits out another number based on a specific rule. Our rule here is 2x³ + 2x² - 7x + 5. This means whatever number we plug in for x, we're going to cube it, square it, multiply it by 7, and then add and subtract these results according to the formula. Easy peasy!

The Magic of Substitution

Now comes the fun part – substitution! We need to find g(0), which basically means we're replacing every x in the function with 0. So, our equation transforms from g(x) = 2x³ + 2x² - 7x + 5 to g(0) = 2(0)³ + 2(0)² - 7(0) + 5. See how we just swapped the x with a 0? That's the core of function evaluation.

The Power of Zero

Zero is a magical number in math, especially when it comes to multiplication. Any number multiplied by zero is zero. This makes our lives super easy! Let's simplify our equation step by step:

• 2(0)³ = 2 * 0 = 0
• 2(0)² = 2 * 0 = 0
• -7(0) = 0

So, our equation g(0) = 2(0)³ + 2(0)² - 7(0) + 5 now looks like g(0) = 0 + 0 - 0 + 5. We've eliminated all those pesky terms with x, thanks to the power of zero!

The Grand Finale: Addition and Subtraction

We're almost there! Now we just need to do the simple addition and subtraction. g(0) = 0 + 0 - 0 + 5 simplifies directly to g(0) = 5. Boom! We've found our answer.

Choosing the Correct Option

Looking at our options, A is 3, B is 5, C is 6, and D is 7. Our calculated value of g(0) is 5, which perfectly matches option B. So, B is our winner!

Key Takeaways for Function Evaluation

• Understand the Function: Get what the function is asking you to do. It's a rule that transforms input (x) into output (g(x)).
• Substitute Carefully: Replace every x with the given value. Double-check your substitution to avoid mistakes.
• Simplify Step-by-Step: Break down the equation into manageable steps. Follow the order of operations (PEMDAS/BODMAS).
• The Magic of Zero: Remember that anything multiplied by zero is zero. This can significantly simplify your calculations.

Q4. Cracking the Code of Arithmetic Sequences

Alright, let's move on to our second challenge: arithmetic sequences. These are sequences where the difference between consecutive terms is constant. Think of it like a staircase where each step is the same height. Our sequence is 3 + 9 + 15 + ... + 129, and we need to find out how many terms are in it. Exciting stuff!

Deciphering Arithmetic Sequences

So, what exactly is an arithmetic sequence? Imagine you're counting, but instead of going up by one each time, you're going up by the same amount – that's an arithmetic sequence! Each number in the sequence is called a term. The difference between consecutive terms is called the common difference. This common difference is the key to unlocking these sequences.

Finding the Common Difference

First things first, we need to figure out the common difference in our sequence 3 + 9 + 15 + ... + 129. To do this, we simply subtract any term from the term that follows it. Let's subtract the first term (3) from the second term (9): 9 - 3 = 6. So, our common difference is 6. This means we're adding 6 to each term to get the next term in the sequence. Cool, right?

The Nth Term Formula: Your Secret Weapon

Now comes the formula that will save the day – the formula for the nth term of an arithmetic sequence. This formula lets us find any term in the sequence without having to list them all out. It looks like this:

• aₙ = a₁ + (n - 1)d

Where:

• aₙ is the nth term (the term we want to find)
• a₁ is the first term in the sequence
• n is the term number (what we're trying to find)
• d is the common difference

Don't let the letters scare you! It's just a tool to help us solve the problem.

Applying the Formula to Our Problem

In our case, we know the last term in the sequence is 129. This is our aₙ. We also know the first term (a₁) is 3, and the common difference (d) is 6. What we don't know is n, the number of terms in the sequence. That's what we're trying to find! Let's plug our known values into the formula:

• 129 = 3 + (n - 1)6

Now we have an equation with one unknown (n). Time to put our algebra skills to work!

Solving for N: A Little Bit of Algebra

Let's solve for n step by step:

1. Subtract 3 from both sides: 129 - 3 = (n - 1)6, which simplifies to 126 = (n - 1)6
2. Divide both sides by 6: 126 / 6 = n - 1, which simplifies to 21 = n - 1
3. Add 1 to both sides: 21 + 1 = n, which gives us n = 22

Ta-da! We've found that n is 22. This means there are 22 terms in our arithmetic sequence.

Picking the Right Choice

Looking at our options, A is 33, B is 22, C is 11, and D is 10. Our calculated value of n is 22, which corresponds perfectly with option B. So, B is the correct answer!

Key Takeaways for Arithmetic Sequences

• Common Difference is Key: Find the constant difference between terms.
• Nth Term Formula: aₙ = a₁ + (n - 1)d is your best friend.
• Plug and Play: Substitute known values into the formula.
• Algebra to the Rescue: Use your algebra skills to solve for the unknown.

Wrapping Up

So, there you have it! We've successfully navigated a function evaluation and an arithmetic sequence problem. Remember, math can be like a puzzle – each piece (concept and formula) fits together to form the solution. Keep practicing, keep exploring, and you'll become a math whiz in no time! You've got this!