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.
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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!