Sequence Formula: Finding Nth Term & Specific Term

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Hey guys! Let's dive into some sequence problems. We're going to figure out how to find the formula for the nth term of a sequence and then use that formula to calculate a specific term, like the 50th or 100th. Let's break it down step by step.

1. Sequence: 40, 24, 12, 4, ...

First, we're given the sequence 40, 24, 12, 4, .... Our mission, should we choose to accept it, is to find a general formula (the nth term) that describes any term in this sequence. Then, we'll use that formula to calculate the 50th term. Let's get started!

Identifying the Pattern

Alright, let's figure out what's going on in this sequence. Is it arithmetic (adding a constant difference) or geometric (multiplying by a constant ratio)? Let's take a look at the differences between consecutive terms:

  • 24 - 40 = -16
  • 12 - 24 = -12
  • 4 - 12 = -8

The differences aren't constant, so it's not an arithmetic sequence. Hmm, let's try ratios:

  • 24 / 40 = 0.6
  • 12 / 24 = 0.5
  • 4 / 12 = 0.333...

The ratios aren't constant either, so it's not a simple geometric sequence. This suggests it's a bit more complex, perhaps involving a quadratic or other type of relationship.

Since the differences between the differences are constant, it indicates a quadratic sequence. Let's analyze this further. The first differences are -16, -12, -8. The second differences are:

  • -12 - (-16) = 4
  • -8 - (-12) = 4

The second difference is constant (4). This confirms that the sequence can be modeled by a quadratic equation of the form:

an^2 + bn + c

Where a, b, and c are constants that we need to determine.

Finding the Coefficients

To find a, b, and c, we can use the first three terms of the sequence. Here’s how:

  • For n = 1: a(1)^2 + b(1) + c = 40 => a + b + c = 40
  • For n = 2: a(2)^2 + b(2) + c = 24 => 4a + 2b + c = 24
  • For n = 3: a(3)^2 + b(3) + c = 12 => 9a + 3b + c = 12

Now we have a system of three equations with three unknowns. Let's solve it.

Subtract the first equation from the second and the second from the third:

  • (4a + 2b + c) - (a + b + c) = 24 - 40 => 3a + b = -16
  • (9a + 3b + c) - (4a + 2b + c) = 12 - 24 => 5a + b = -12

Now, subtract the first of these new equations from the second:

  • (5a + b) - (3a + b) = -12 - (-16) => 2a = 4 => a = 2

Now that we have a, we can find b:

  • 3(2) + b = -16 => 6 + b = -16 => b = -22

Finally, we can find c:

  • 2 - 22 + c = 40 => -20 + c = 40 => c = 60

So, our formula for the nth term is:

Un = 2n^2 - 22n + 60

Calculating the 50th Term

Now that we have our formula, finding the 50th term is easy peasy. Just plug in n = 50:

U50 = 2(50)^2 - 22(50) + 60

U50 = 2(2500) - 1100 + 60

U50 = 5000 - 1100 + 60

U50 = 3900 + 60

U50 = 3960

So, the 50th term of the sequence is 3960.

2. Sequence: 10, 6, 1, -4, ...

Next up, we've got the sequence 10, 6, 1, -4, .... Same game, different numbers. We want to find the formula for the nth term and then calculate the 100th term. Let's get to it!

Identifying the Pattern

Let's start by looking at the differences between consecutive terms:

  • 6 - 10 = -4
  • 1 - 6 = -5
  • -4 - 1 = -5

Okay, the first difference isn't constant but the difference between the second and third terms are. Let's look at the second differences:

  • -5 - (-4) = -1
  • -5 - (-5) = 0

Since neither the first nor the second differences are constant, the sequence is not arithmetic or a simple quadratic. However, noting the constant -5 difference between the 3rd and 4th terms hints the sequence may be piecewise or need another approach. Let's calculate a few more terms in the sequence to see if a pattern appears.

To find the general pattern of this sequence, let's assume it can be represented by a polynomial equation of the form:

an^2 + bn + c

We have the first four terms: 10, 6, 1, -4. Now, let's create a system of equations:

  • For n = 1: a(1)^2 + b(1) + c = 10 => a + b + c = 10
  • For n = 2: a(2)^2 + b(2) + c = 6 => 4a + 2b + c = 6
  • For n = 3: a(3)^2 + b(3) + c = 1 => 9a + 3b + c = 1

Subtract the first equation from the second and the second from the third to eliminate c:

  • (4a + 2b + c) - (a + b + c) = 6 - 10 => 3a + b = -4
  • (9a + 3b + c) - (4a + 2b + c) = 1 - 6 => 5a + b = -5

Subtract the first of these new equations from the second:

  • (5a + b) - (3a + b) = -5 - (-4) => 2a = -1 => a = -0.5

Substitute a back into one of the equations to find b:

  • 3(-0.5) + b = -4 => -1.5 + b = -4 => b = -2.5

Substitute a and b back into the first equation to find c:

  • -0.5 - 2.5 + c = 10 => -3 + c = 10 => c = 13

So, the general formula for the nth term is:

Un = -0.5n^2 - 2.5n + 13

Calculating the 100th Term

Now, let's plug in n = 100 to find the 100th term:

U100 = -0.5(100)^2 - 2.5(100) + 13

U100 = -0.5(10000) - 250 + 13

U100 = -5000 - 250 + 13

U100 = -5250 + 13

U100 = -5237

So, the 100th term of the sequence is -5237.

Conclusion

And there you have it! We successfully found the formulas for the nth terms of both sequences and calculated the specified terms. Sequences can be tricky, but breaking them down step by step and identifying the patterns makes it much easier. Keep practicing, and you'll be a sequence master in no time! Hopes this helps, bye!