Calculate the explicit formula, term number 10, and the sum of the first 10 terms for the following geometric series:

2,4,8,16,32

The explicit formula for a geometric series is a

r represents the common ratio between each term below:

r = | a_{n} |

a_{n - 1} |

Looking at all the terms, we see the common ratio (r) is 2, and we have a

We simplify a

ar = 2 x 2 =

Therefore, our explicit formula is

Using our explicit formula with n = 10 and r = 2, we have:

Term # | Math Step 1 | Math Step 2 | Math Step 3 | Term |
---|---|---|---|---|

6 | a_{6} = 2 x 2^{(6 - 1)} | a_{6} = 2 x 2^{5} | a_{6} = 2 x 32 | a_{6} = 64 |

7 | a_{7} = 2 x 2^{(7 - 1)} | a_{7} = 2 x 2^{6} | a_{7} = 2 x 64 | a_{7} = 128 |

8 | a_{8} = 2 x 2^{(8 - 1)} | a_{8} = 2 x 2^{7} | a_{8} = 2 x 128 | a_{8} = 256 |

9 | a_{9} = 2 x 2^{(9 - 1)} | a_{9} = 2 x 2^{8} | a_{9} = 2 x 256 | a_{9} = 512 |

10 | a_{10} = 2 x 2^{(10 - 1)} | a_{10} = 2 x 2^{9} | a_{10} = 2 x 512 | a_{10} = 1024 |

S_{n} = | a_{1}(1 - r^{n}) |

1 - r |

With n = 10, and r = 2, we get:

S_{10} = | 2(1 - 2^{10}) |

1 - 2 |

S_{10} = | 2(1 - 1024) |

-1 |

S_{10} = | 2(-1023) |

-1 |

S_{10} = | -2046 |

-1 |

S