Josh has a itunes gift card worth $75. After he purchases the first song on itunes, the cards value is $74.11. After he purchases the second song, its value is $73.22. After he purchases the third song, its value is $72.33. Assuming the pattern continues, write an equation to define A(n), the amount of money on the gift card after n song purchases. How many songs can Josh purchase with this card only?

Step-by-step explanation:

The worth of the card =$75

The value of the card decreases after each music purchase forming a sequence:

75,74.11,73.22,72.33...

Now,

We see that the sequence forms an arithmetic sequence:

Where the first term, a is 75

Common difference, d=-0.89

We want to write an equation to define A(n), the amount of money on the gift card after n song purchases.

For an arithmetic sequence,the nth term A(n)=a+(n-1)d

Therefore:

The amount of money on the gift card after n song purchases.

A(n)=75-0.89(n-1)

A(n)=75+0.89-0.89n

A(n)=75.89-0.89n

To find out how many songs can Josh purchase with this card only, we take our nth term A(n) of the sequence to be 0 and solve for n.

A(n)=75.89-0.89n

0=75.89-0.89n

0.89n=75.89

Divide both sides by 0.89

n=85.3

n=85

Therefore, Josh can purchase 85 songs with this card only