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The sum of the first 10 terms of an arithmetic sequence is 530. What is the first term if the last term is 80? What is the common difference?

Sagot :

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Arithmetic Sequence:  

The first term of the sequence is 26 and the common difference is 6.  

Definition:

       An arithmetic sequence is a sequence of numbers such that the difference of any two succeeding terms of the sequence is a constant while an arithmetic series is the sum of an arithmetic sequence. We find the sum by adding the first, a and last term, an, divide by 2 in order to get the mean of the two values and then multiply by the number of values.

Step by Step Solution:  

  • Write the given.  

       S = 530 (sum of the first 10 terms of the arithmetic sequence)  

       a = 80 (10th term of the arithmetic sequence)  

  • Identify what is asked.  

       a = ____ (first term of the arithmetic sequence)  

       d = ____ (common difference of the arithmetic sequence)  

  • Identify the formula to be used.  

       a = a + (n – 1)d  

       S = [2a + (n – 1)d]n/2  

  • Write the solution.  

       a = a + (10 – 1)d  

       80 = a + 9d  

  • Rename the equation to a by adding -9d to both sides of the equation.  Thus, 80 + (-9d) = a + 9d + (-9d) which when simplified is 80 – 9d = a  
  • Substitute the derived equation for a in the equation for the sum of  the arithmetic sequence to solve for the common difference.

       S = [2a + (n – 1)d]n/2  

       530 = [2(80 – 9d) + (10 – 1)d]10/2  

       530 = [160 – 18d + 9d]5  

       530 = [160 – 9d]5  

       530 = 800 – 45d  

  • Combine the similar terms.

       530 + (-800) = – 45d  

  • Divide both sides by -45 to solve for d.  

       6 = d . Therefore, the common difference is 6.  

  • Now, find the first term of the arithmetic sequence using the formula:  80 – 9d = a  

       80 – 9(6) = a  

       80 – 54 = a  

       26 = a . Therefore, the first term of the arithmetic sequence is 26.  

For more information regarding arithmetic sequence, read the following links:

https://brainly.ph/question/825249

https://brainly.ph/question/600653

https://brainly.ph/question/651839

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