Question
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
.
2.2.1
Determine the general term of the pattern in the form
ION 3
If
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:
-
For
: -
For
: -
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
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!