Pregunta
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20 The sum of the first 53 terms of an
arithmetic series is 4240 , while the
seventh term is equal to 20 .
Find the first term and the common
difference.

Ask by Ellis Page. in South Africa
Jan 30,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

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The first term is 2 and the common difference is 3.

Solución

To find the first term ( ) and the common difference ( ) of the arithmetic series, we’ll use the given information:
  1. Sum of the first 53 terms ( ): 4240
  2. Seventh term ( ): 20

Step 1: Use the Sum Formula

The sum of the first terms of an arithmetic series is given by:
Plugging in the values:
Simplify:
Divide the entire equation by 2:

Step 2: Use the Formula for the -th Term

The -th term of an arithmetic series is:
For the 7th term:

Step 3: Solve the System of Equations

From Equation 2:
Substitute from Equation 2 into Equation 1:
Now, substitute back into Equation 2:

Answer:

  • First term ( ) = 2
  • Common difference ( ) = 3

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Extra Insights

To find the first term and the common difference of the arithmetic series, we can use the formulas related to the sum of an arithmetic series and the nth term.
The formula for the sum of the first terms of an arithmetic series is:
Where is the sum of the first terms, is the first term, is the common difference, and is the number of terms.
Given:
We can plug in these values:
Multiplying both sides by 2:
Now divide both sides by 53:
It is also given that the seventh term is 20:
Now we have two equations:
From equation (2), we can express :
Now substitute equation (3) into equation (1):
This simplifies to:
Combining like terms gives:
Subtracting 40 from both sides:
Now divide by 40:
Now that we have , substitute it back into equation (3) to find :
Thus, the first term is 2, and the common difference is 3.
Final Answer: The first term is 2, and the common difference is 3.

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