Chapter ISG, Problem 3.6.7P

a.

To determine

To write a formula for the n^th term of the given descriptions

Answer to Problem 3.6.7P

The expression for the n^th term for the given description is, $a_n=-a-d(n-1)$

Explanation of Solution

Given information:

First term (a1) is negative

Common difference (d) is also negative

Formula Used:

Arithmetic sequence formula is,

  $a_n=a1+(n-1)d$

Where, a_n is the n^th term

a1 is the first term of the sequence

d is the common difference.

Calculation:

Let the first term (a1) be “-a ”

And the common difference (d) be “-d ”

Substituting the value in the arithmetic sequence formula,

  $a_n=(-a)+(n-1)(-d)$

Opening the brackets,

  $a_n=-a-d(n-1)$

Conclusion:The expression for the n^th term for the given description is, $a_n=-a-d(n-1)$

b.

To determine

To write a formula for the n^th term of the given descriptions

Answer to Problem 3.6.7P

The formula for the n^th term of the given descriptions is, $a_n=7(n-1)-12$

Explanation of Solution

Given information:Second term is -5

Common difference (d) is 7

Formula Used:

Arithmetic sequence formula is,

  $a_n=a1+(n-1)d$

Where, a_n is the n^th term

a1 is the first term of the sequence

d is the common difference.

Calculation:

For the first term, substituting the values in arithmetic sequence,

  $-5=a1+(2-1)7$

So,

  $-5=a1+7$

Shifting left hand side values to right hand side,

  $a1=-5-7$

On solving,

  $a1=-12$

Hence, first term of the sequence is -12

For the formula of the n^th term, substituting the values,

  $a_n=-12+(n-1)7$

On re-arranging the variables,

  $a_n=7(n-1)-12$

Conclusion:The formula for the n^th term of the given descriptions is, $a_n=7(n-1)-12$

c.

To determine

To write a formula for the n^th term of the given descriptions

Answer to Problem 3.6.7P

The formula for the n^th term of the given descriptions is, $a_n=10-2(n-1)$

Explanation of Solution

Given information: Second term is 8

Third term is 6

Formula Used:

Arithmetic sequence formula is,

  $a_n=a1+(n-1)d$

Where, a_n is the n^th term

a1 is the first term of the sequence

d is the common difference.

Calculation:

For the common difference (d),

Subtracting third and second term,

  $d=6-8$

On solving,

  $d=-2$

Hence, common difference is -2

For the first term, substituting the values in arithmetic sequence,

  $8=a1+(2-1)(-2)$

So,

  $8=a1-2$

Shifting left hand side values to right hand side,

  $a1=8+2$

On solving,

  $a1=10$

Hence, first term of the sequence is 10

For the formula of the n^th term, substituting the values,

  $a_n=10+(n-1)(-2)$

On solving,

  $a_n=10-2(n-1)$

Conclusion: The formula for the n^th term of the given descriptions is, $a_n=10-2(n-1)$