Question

Establish the following by mathematical induction

a) Σ_{i=1}^{n} i(2^i)=2+(n-1)2^(n+1) b) Σ_{i=1}^{n} 2(3^(i-1))=3^n-1 c) Σ_{i=1}^{n} i/
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Answer 1

The question involves proving summation formulas using mathematical induction, including a specific formula involving a sum of terms each multiplied by a power of two.

Step-by-step explanation:

The student has asked a question related to mathematical induction and a series of summations. Let's focus on part a). We need to show by induction that the sum from i=1 to n of i(2^i) is equal to 2 + (n - 1)2^(n + 1).

1. First, we establish the base case where n=1. We need to show that the statement holds.
2. Next, assuming the statement is true for n=k, we show that it must also be true for n=k+1.
3. By successfully demonstrating both steps, we confirm the formula by mathematical induction.

We would use a similar process for part b) and part c) but with their respective formulas.