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23. Critical Thinking Lucia knows the fourth term in a sequence is 55 and the ninth term in the same sequence is 90. Explain how she can find the common difference for the sequence. Then use the common difference to find the second term of the sequence.

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DeanR

Let's call a the first term and d the common difference.

[tex]a_n = a+(n-1)d[/tex]

The fourth term is 55:

[tex]a_4 = 55 = a + 3d[/tex]

The ninth term is 90

[tex]a_9 = 90 = a + 8d[/tex]

Subtracting,

[tex]90 - 55 = 8d - 3d[/tex]

[tex]35 = 5d[/tex]

[tex]d = 7[/tex]

Answer: common difference of 7

We found the common difference.  Substituting,

[tex]a = 55 - 3d = 55 - 21 = 34[/tex]

Second term

[tex]a_2 = a + d = 34 + 7 = 41[/tex]

Answer: second term of 41


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