An explained answer
1 answer:
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
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as you can see, the terms exponents, for the first term, starts at highest, 5 in this case, then every element it goes down by 1, till it gets to 0
for the second term, starts at 0, and every element it goes up by 1, till it gets to the highest
now, to get the coefficient, they way I get it, is "the product of the current coefficient and the exponent of the first term, divided by the exponent of the second term plus 1"
notice the first coefficient is always 1
so...how did we get 10 for the 3rd element? well, 5*4/2
how did we get 10 for the fourth element? well, 10*2/4
and from there, you can simplify the elements of the expansion by combining the coefficients
like for example, the 7th element of (3a+4b)⁸ will then be 1032192a²b⁶