Question
A quadratic sequence is defined with the following properties:
2.3 .1 Write down the value of:
(a)
(b)
2.3 .2 Calculate the value of
2.3 .1 Write down the value of:
(a)
(b)
2.3 .2 Calculate the value of
Ask by Warner Simpson. in South Africa
Jan 23,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
2.3.1
- (a)
- (b)
2.3.2
Solution
Let’s analyze the quadratic sequence step by step.
2.3.1 Differences in the Sequence
A quadratic sequence has the general form:
The first differences (
Given:
(a)
Observe that the first differences increase by 6 each time:
Answer:
(b)
Using the arithmetic pattern:
So,
Answer:
2.3.2 Calculating
We know:
To find a general formula for
Calculating the sum:
Thus, the general formula is:
Given
Now, using
Calculate
Answer:
Summary:
- 2.3.1
- (a)
- (b)
- (a)
- 2.3.2
-
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Simplify this solution Extra Insights
The differences between consecutive terms in a quadratic sequence form an arithmetic sequence. Based on the given differences:
- We have the differences:
, , and . - The difference increases by 6 each time, indicating
(which is 19 + 6).
For the second part, since the difference pattern is predictable (increases by 6), the difference for
Now let’s compute
- The difference
. -
.
Given
Continuing backwards, we can find
-
- Working our way up through the differences, we can eventually derive
.
Ultimately,
