Question
Given the geometric series
, where
is the first tem and
is the common ratio. Prove thar the sum to
terms of this series is given by
The first two termbs of a geometric sequence with constant ratio
, and an arithmetic sequence with constant first difference,
, is tise same. The first term is 12.
4.2.1 Write down the second and third terns of EACH sequence in terms of
and
.
4.2.1 Write down the second and third terns of EACH sequence in terms of
Ask by Barnett Schofield. in South Africa
Jan 23,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The second term of the geometric sequence is
and the third term is
. The second term of the arithmetic sequence is
and the third term is
.
Solution

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The Deep Dive
To solve this problem, let’s break it down step by step!
For the geometric series:
The sum of the first terms can be derived by multiplying the series by the common ratio
and then subtracting from the original series. This gives:
The sum of the first
Then multiply by
:
Now subtract these two equations:
Factoring out
gives:
Thus, we can express
as:
Rearranging shows it as
when
.
Now, for the sequences:
In the arithmetic sequence, if the first term is , then:
In the arithmetic sequence, if the first term is
- Second term:
- Third term:
In the geometric sequence, the first term is also
:
- Second term:
- Third term:
So, we have:
- Arithmetic sequence: Second term is
, Third term is . - Geometric sequence: Second term is
, Third term is .