Question
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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 .

Ask by Barnett Schofield. in South Africa
Jan 23,2025

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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 .

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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:
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:
  • 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 .

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