The sum of the first `n` terms is given by

`S_n = frac(n(1_1 + a_n))(2)`

which is the average of the first and last terms multiplied by the number of terms.

What is the sum of the first 50 terms of the sequence that begins 8, 11, 14, 17, 20, ...

8, 11, 14, 17, 20, …

The common difference is 3. The 0^{th} term is 8 - 3 = 5

The `nth` term is `3n + 5`

The 50th term is 3 x 50 + 5 = 155

The sum of a sequence is `S_n` | = `frac(n(a_1 + a_n))(2)` |

= `frac(50(5 + 155))(2)` | |

= `4075` |

Answer: 4075

A sequence begins 7, 11, 15, 19, … The sum of the sequence is 2208. How many terms are in the sequence?

The common difference is 4. The zeroth term is 7 - 4 = 3.

The `n`th term is given by 4n + 3.

Sum of a sequence is | `S_n` | `=` | `frac(n(a_1 + a_n))(2)` |

`2208` | `=` | `frac(n(7 + a_n))(2)` | |

Substitute `4n+3` for `a_n` | `2208` | `=` | `frac(n(7 + (4n + 3)))(2)` |

`4416` | `=` | `10n + 4n^2` | |

`0` | `=` | `4n^2 + 10n + 4416` | |

Solve for n | `n` | `=` | `frac(-10 ± 266)(8)` |

Only positive needed | `n` | `=` | `32` |

Answer: 32

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