I need help with this math problem please.
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Answer:
Fibonacci Series has been explained and the general term and shortcut to find the corresponding term has been attached
Step-by-step explanation:
Fibonnaci is a beautiful series in mathematics where the term in the series is the sum of the previous two terms of the corresponding term in the series.
Its general form is denoted by
,
where represents the
of the Fibonnaci series.
The special thing about the Fibonacci series is that the more number of terms we proceed the ratio of the two consecutive term comes closer to the value of the Golden Ratio(φ) whose value is 1.618034.
But there is another method to find out the terms of the the Fibonacci series, which takes into account the value of φ. The formula for the following is as follows
The last three terms of the given binomial expansion are 40x²y³ + 10xy⁴ + y⁵
<h3>Binomial expansion</h3>From the question, we are to determine the last three terms of the binomial expansion 32x⁵ + 80x⁴y + 80x³y² + __ + __ + __
The highest power of the expansion is 5
Using the binomial expansion theorem
(m +n)⁵ = m⁵ + 5m⁴n + 10m³n² + 10m²n³ + 5mn⁴ + n⁵
Now, we will determine the values of m and n by comparison
By comparison,
m⁵ = 32x⁵
m⁵ = (2x)⁵
∴ m = 2x
Also,
5m⁴n = 80x⁴y
Divide both sides by 5, to get
m⁴n = 16x⁴y
Put m = 2x into the equation
(2x)⁴n = 16x⁴y
16x⁴n = 16x⁴y
∴ n = y
Thus, the expression being expanded is (2x + y)⁵
By binomial expansion, the expansion of (2x + y)⁵ is
(2x + y)⁵ = (2x)⁵ + 5(2x)⁴y + 10(2x)³(y)² + 10(2x)²(y)³ + 5(2x)(y)⁴ + y⁵
(2x + y)⁵ = 32x⁵ + (5×16)x⁴y + (10×8)x³(y)² + (10×4)x²(y)³ + (5×2)x(y)⁴ + y⁵
(2x + y)⁵ = 32x⁵ + 80x⁴y + 80x³y² + 40x²y³ + 10xy⁴ + y⁵
Hence, the last three terms of the given binomial expansion are 40x²y³ + 10xy⁴ + y⁵
Learn more on Binomial expansion here: brainly.com/question/13602562
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