Pregunta
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Consider the arithmetic sequence and the geometric sequence .
Which sequence has the larger value at position 13 ? What is the difference between the larger
and smaller value, correct to two decimal places?

Ask by Cox Barnett. in Australia
Mar 29,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

Responder

The geometric sequence has a larger 13th term, and the difference between the two sequences at that position is approximately 133.49.

Solución

We first find the 13th term of the arithmetic sequence and the geometric sequence.
Arithmetic Sequence:
The sequence is given by
For :
Geometric Sequence:
The sequence is given by
For :
We now compute . Using a calculator or by successive multiplication, we find:
Thus,
Comparison and Difference:
The arithmetic sequence has a 13th term of and the geometric sequence approximately . Therefore, the geometric sequence has the larger 13th term.
The difference is:
Rounded to two decimal places, the difference is approximately:

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Mind Expander

To find the values at position 13 for both sequences, let’s first analyze each one.
For the arithmetic sequence :
  • The first term and the common difference .
  • The nth term formula for an arithmetic sequence is given by:
  • For :
Now for the geometric sequence :
  • The first term and the common ratio .
  • The nth term formula for a geometric sequence is given by:
  • For :
Now we calculate :
So:
Now we compare the two values:
Clearly, the geometric sequence has the larger value.
To find the difference:
Thus, the larger value is from the geometric sequence and the difference between the larger and smaller value at position 13 is:

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