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Given a Geometric Progression (GP): (128)/(125), ..., 100, 250, 625. Find the number of terms and the first three terms.
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3 partsStep-by-step solution
10 stepsPart 7
Given a Geometric Progression (GP): .
Part a
Find the number of terms.
Identify the first term and common ratio.
The first term is .
From the given terms , we can find the common ratio .We can verify this with the next pair:
Use the formula for the -th term of a GP.
The last term is . The formula for the -th term is .
Substitute the values of , , and :Solve for .
Express the numbers as powers of their prime factors: , , .
For this equality to hold, the powers of the bases must match.
Equating the powers of 2:Equating the powers of 5:
Both equations yield .
The number of terms is .
Part b
Write down the first three terms.
Identify the first term.
The first term is .
Calculate the second term.
The second term is .
Calculate the third term.
The third term is .
The first three terms are .
Part 8
Which term of the sequence: is ?
Convert mixed numbers to improper fractions and identify the first term.
The sequence is
The first term is .Find the common ratio .
Verify with the next pair:
The common ratio is .
Use the formula for the -th term of a GP and solve for .
We want to find such that .
The formula is .Express numbers as powers of their prime factors and solve for .
Substitute these into the equation:Equating the powers of 2:
Equating the powers of 3:
Both equations give .
The term is the .
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