Problem 5 Using the Geometric Series Formula

According to the story on the preceding page, how many total kernels of wheat did the soldier request?

• Step 1 Identify the first term, common ratio, and the number of terms.

  a 1 = 1 , r = 2 , n = 64 eh sub 1 , equals 1 . comma , r equals 2 , comma , n equals 64

• Step 2 Use the sum formula.

  S n = a 1 ( 1 − r n ) 1 − r Write the sum formula . S 64 = 1 ( 1 − 2 64 ) 1 − 2 Substitute for  a 1 , r , and  n .   = 2 64 − 1 ≈ 1.845 × 10 19 Simplify . table with 3 rows and 3 columns , row1 column 1 , s sub n , column 2 equals . fraction eh sub 1 . open . 1 minus , r to the n . close , over 1 minus r end fraction , column 3 cap writethesumformula . . , row2 column 1 , s sub 6 4 , column 2 equals . fraction 1 . open . 1 minus , 2 to the sixty fourth . close , over 1 minus 2 end fraction , column 3 cap substitutefor . eh sub 1 , comma r comma , and , n . , row3 column 1 , , column 2 equals , 2 to the sixty fourth , minus 1 almost equal to , 1.845 , times , 10 to the nineteenth , column 3 cap simplify , . , end table

  There will be approximately 1.845 × 10 19 1.845 , times , 10 to the nineteenth  kernels of wheat.

✓ Got It?

2. To save money for a vacation, you set aside $100. For each month thereafter, you plan to set aside 10% more than the previous month. How much money will you save in 12 months?

Take Note Key Concept: Infinite Geometric Series

An infinite geometric series with first term a 1 eh sub 1  and common ratio | r | < 1 vertical line r vertical line less than 1  has a finite sum

S = a 1 1 − r . s equals . fraction eh sub 1 , over 1 minus r end fraction . .

An infinite geometric series with | r | ≥ 1 vertical line r vertical line greater than or equal to 1  does not have a finite sum.