Geometry given right triangle ABC with altitude BD drawn to hvpotenuse AC. If AD = 8
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Answer:
225
Step-by-step explanation:
When you fill in values of n, you find the series is an arithmetic series of 15 terms with a first term of 1 and a common difference of 2. The formula for the sum of such a series can be used.
<h3>Terms</h3>Looking at terms of the series for different values of n, we find ...
for n = 1: 2(1) -1 = 1 . . . . . the first term
for n = 2: 2(2) -1 = 3 . . . . the second term; differs by 3-1=2
for n = 15: 2(15) -1 = 29 . . . . the last of the 15 terms
<h3>Sum</h3>The sum of the terms of an arithmetic series is the product of the average term and the number of terms. The average term is the average of the first and last terms.
Sum = (1 +29)/2 × 15 . . . . . . average term × number of terms
Sum = 15 × 15 = 225
The sum of the series is 225.
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<em>Additional comment</em>
Based on the first term (a1), the common difference (d), and the number of terms (n), the sum can also be written ...
S = (2×a1 +d(n -1))(n/2)
For the parameters of this series, the sum is ...
S = (2(1) +2(15 -1))(15/2) = 30(15/2) = 225
