Find the number of terms in the following geometric series: 100 + 99 + 98.01 + ... + 36.97

Question

exalantaswo · Accepted Answer

A geometric series is a series of the form            a      0        +          a      0        r    +          a      0              r      2        +          a      0              r      3        +    ...  where             a      0       is the initial term and r is the common ratio between terms. We can easily find r by dividing any term after the first by the prior term. So in this case we have            a      0        =    100   and       r    =          99      100        =    0.99  Now, looking at the general form of the series, we can see that the             n              th             term has the form             a      0              r              n        -        1            . Thus, as we have the last term in the series, we simply need to solve for n for that term to find the total number of terms.      36.97    =          a      0              r              n        -        1              =    100                  (        0.99        )                    n        -        1                  ⇒    0.3697    =          0.99              n        -        1                  ⇒          ln              (        0.3697        )              =          ln              (                  0.99                      n            -            1                          )              =          (      n      -      1      )              ln              (        0.99        )                  ⇒    n    =                  ln                  (          0.3697          )                                      ln                      (            0.99            )                                +    1    ≈    100  Thus there are 100 terms in the series.

Find the number of terms in the following geometric series: 100 + 99 + 98.01 + ... + 36.97

Answered question

Answer & Explanation

New Questions in Algebra I