Question
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Given the quadratic number pattern:
2.2.1
Determine the general term of the pattern in the form .


ION 3
If , determine the value of .

Ask by Sandoval Medina. in South Africa
Jan 26,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The general term of the pattern is . The 50th term is 14,550. The sum of the first first-differences is . To reach a term of 21,060, 20 consecutive first-differences were added to the first term.

Solution

Let’s solve the problem step by step.

2.2.1 Determine the general term of the pattern in the form .

We have the sequence:
Let’s denote the terms as follows:
We need to find such that:
We can set up a system of equations using the known terms:
  1. For :
  2. For :
  3. For :
Now we have the following system of equations:
Next, we can eliminate by subtracting equation (1) from equations (2) and (3):
From (2) - (1):
From (3) - (1):
Now we have a new system of equations:
Subtract equation (4) from (5):
Substituting back into equation (4):
Now substituting and back into equation (1):
Thus, the general term is:

2.2.2 Calculate the value of the term of the pattern.

Now we can calculate :
Calculating :
Now substituting:

2.2.3 Show that the sum of the first first-differences of this pattern can be given by .

The first differences can be calculated as:
Calculating :
Expanding :
Thus,
Now, the sum of the first first-differences:
This can be simplified:

2.2.4 How many consecutive first-differences were added to the first term of the quadratic number pattern to obtain a term in the quadratic number pattern that has a value of ?

We need to find such that:
Using the general term:
Rearranging gives:
Dividing the entire equation by 3:
Now we can solve this quadratic equation using the quadratic formula:
Calculating the discriminant:
[
b^2 - 4ac = 9 + 4 \cdot

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Beyond the Answer

To determine the general term of the pattern , we start with the given values of the sequence. Plugging in the first few terms gives us a system of equations to solve for , , and . After working through that, you find that .
Now for the 50th term, substitute into the determined general formula. You would calculate , which simplifies down to after doing the math.
For the sum of the first first differences, if the first differences are noted, their sum can indeed be expressed as . Verifying the pattern from the actual differences and cumulating them will lead to this quadratic form.
If you seek in terms of how many consecutive first differences were added to the first term, you would derive how many terms fit into using the equation already established after substituting and solving for .
Now, let’s dive into the infinite series sum! With , we can rewrite it as starting from . Using the sum formula for a geometric series, you find corresponds to the value that, when plugged into the sum formula, gives you . The tricky math reveals that equals . Happy calculating!

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