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Question 5
A geometric series has first term a and common ratio r. The second term of the series is 4 and the sum to infinity of the series is 25. (a) Show that $2.5r^2 - 2.5r... show full transcript
Step 1
Answer
To find the relationship between the first term (a) and the common ratio (r), we set up the equation for the second term:
Using the sum to infinity for the series, we have:
From the equations, substituting for a gives:
Now substituting into the equation :
Which simplifies to:
Rearranging gives:
Thus,
.
Step 2
Step 3
Step 4
Answer
The formula for the sum of the first n terms in a geometric series is derived from:
Factoring gives us:
Using the formula for the sum of a geometric series:
This proves the required formula.
Step 5
Answer
Using the formula for S_n, we substitute the larger value of r, which is 5, and the corresponding value of a, which is 0.8:
Solving for when this exceeds 24, we have:
Which simplifies to:
Therefore,
Rearranging gives us:
Upon calculating:
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1.1 Proof
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1.2 Proof by Contradiction
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2.1 Laws of Indices & Surds
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2.2 Quadratics
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2.3 Simultaneous Equations
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2.4 Inequalities
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2.5 Polynomials
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2.6 Rational Expressions
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2.7 Graphs of Functions
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2.8 Functions
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2.9 Transformations of Functions
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2.10 Combinations of Transformations
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2.11 Partial Fractions
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2.12 Modelling with Functions
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2.13 Further Modelling with Functions
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3.1 Equation of a Straight Line
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3.2 Circles
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4.1 Binomial Expansion
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4.2 General Binomial Expansion
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4.3 Arithmetic Sequences & Series
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4.4 Geometric Sequences & Series
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4.5 Sequences & Series
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4.6 Modelling with Sequences & Series
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5.1 Basic Trigonometry
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5.2 Trigonometric Functions
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5.3 Trigonometric Equations
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5.4 Radian Measure
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5.5 Reciprocal & Inverse Trigonometric Functions
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5.6 Compound & Double Angle Formulae
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5.7 Further Trigonometric Equations
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5.8 Trigonometric Proof
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5.9 Modelling with Trigonometric Functions
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6.1 Exponential & Logarithms
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6.2 Laws of Logarithms
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6.3 Modelling with Exponentials & Logarithms
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7.1 Differentiation
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7.2 Applications of Differentiation
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7.3 Further Differentiation
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7.4 Further Applications of Differentiation
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7.5 Implicit Differentiation
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8.1 Integration
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8.2 Further Integration
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8.3 Differential Equations
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9.1 Parametric Equations
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10.1 Solving Equations
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10.2 Modelling involving Numerical Methods
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11.1 Vectors in 2 Dimensions
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11.2 Vectors in 3 Dimensions
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