When finding the nth term in an arithmetic sequence, how do I solve a sequence with fractions? For example {−445, −135, 135, 445⋯} When 445 is the 4th term what would the seventh term be and why is the final answer 725?

Question

When finding the nth term in an arithmetic sequence, how do I solve a sequence with fractions? For example {−445, −135, 135, 445⋯} When 445 is the 4th term what would the seventh term be and why is the final answer 725?

Cullen · Accepted Answer

Step 1  To justify that the seventh term of the given arithmetic sequence is 725 (given)  Step 2  To start with, an arithmetic sequence is a sequence of the form {a, a+d, a+2d,⋯}, where a and d are any numbers (integers, fractions, real numbers, positive, negative). There is no restriction on the type of numbers a and d.  Step 3  All the standard formulae hold for an arithmetic sequence even if the terms are fractions. (a is the first term and d is the common difference with the usual notation)  Given an arithmetic sequence (of any type of numbers) {a, a+d, a+2d,⋯}  nth term=a+(n−1)d  Step 4  The first term a and the common difference d are identified  Given the sequence  {−445, −135, 135, 445,⋯} same as  {−245, −85, 85, 245,⋯}  We see that this is an arithmetic sequence with a=−245 and d=165  Step 5  We can determine the 7th term either directly or using the formula for the nth term. First we find the 7th term directly  {−245, −85, 85, 245,⋯}  5th term=245+165=8  (displaystyletext{6th term=8+frac{16}{leftlbrace{5}rightrbrace}={frac{{{56}}}{{{5}}}})  7th term=565+165=725 (as given)  Step 6  ANSWER: Finding the 7th term using the formula for the nth term  {−245, −85, 85, 245,⋯}  a=−245, d=165;  ntn term=a+(n−1)d  So, 7th term=−245+6×165  =−24+965  =725 (as required)

When finding the nth term in an arithmetic sequence, how do I solve a sequence w

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