Find the sum of the first 41 terms of the following series, to the nearest integer.
1 answer:
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Answer:
see explanation
Step-by-step explanation:
Expand both factors and collect like term
Using Pascal' triangle with n = 6 to obtain the coefficients
1 6 15 20 15 6 1
Decreasing powers of 1 from to
Increasing powers of 3x from to
= 1. + 6.
+ 15.
+ 20.
+ 15.1²
+ 6.
+ 1.
= 1 + 18x + 135x² + 540x³ + 1215 + 1458
+ 729
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= 1. + 6.
+ 15.
+ 20.
+ 15.1²
+ 6.
+ 1.
= 1 - 18x + 135x² - 540x³ + 1215 - 1458
+ 729
----------------------------------------------------------------------------------
Collecting like terms from both expressions
+
= 2 + 270x² + 2430 + 1458
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(2)
Using Pascal's triangle with n = 5
1 5 10 10 5 1
Decreasing powers of 1 from to
Increasing powers of 2x from to
= 1. + 5.
+ 10.
+ 10.
+ 5.
+ 1.
= 1 + 10x + 40x² + 80x³ + 80 + 32