\(\displaystyle \text{Prove or disprove the statement }\)
\(\displaystyle \sum\limits_{i = 1}^{n + 1} {(i2^i )} = n2^{n + 2} + 2,\forall \text{ integers n} \geqslant \text{0}\)
\(\displaystyle \text{Step 1:}\)
\(\displaystyle \text{Show that p(a) is true ['a' being the minimum value of p(x), or 1)}\)
\(\displaystyle \sum\limits_{i = 1}^{(1) + 1} {(i2^i )} = (1)2^{(1) + 2} + 2\)
\(\displaystyle (1*2^1 ) + (2*2^2 ) = 10\)
\(\displaystyle 2 + 8 = 10\)
\(\displaystyle 10 = 10\text{ [true for p(a)]}\)
\(\displaystyle \text{Step 2:}\)
\(\displaystyle \text{Suppose that p(k) holds for all k} \geqslant \text{a}\)
\(\displaystyle \sum\limits_{i = 1}^{k + 1} {(i2^i )} = k2^{k + 2} + 2\)
\(\displaystyle \text{I start from the left hand side using the difinition of}\sum {}\)
\(\displaystyle \sum\limits_{i = 1}^{(k + 1) + 1} {(i2^i )} = \sum\limits_{i = 1}^{k + 2} {(i2^i )} = (1)2^1 + (2)2^2 + (3)2^3 ...(k)2^k + (k + 1)2^{k + 1} + (k + 2)2^{k + 2}\)
\(\displaystyle \text{ = }\sum\limits_{i = 1}^{k + 1} {(i2^i )} \text{ + }(k + 2)2^{k + 2}\)
\(\displaystyle \text{Substitute for induction hypothesis}\)
\(\displaystyle \sum\limits_{i = 1}^{k + 1} {(i2^i )} = k2^{k + 2} + (2 + (k + 2)2^{k + 2} )\)
\(\displaystyle \text{Where do I go from here}...\text{Am I correct so far?}\)
\(\displaystyle \sum\limits_{i = 1}^{n + 1} {(i2^i )} = n2^{n + 2} + 2,\forall \text{ integers n} \geqslant \text{0}\)
\(\displaystyle \text{Step 1:}\)
\(\displaystyle \text{Show that p(a) is true ['a' being the minimum value of p(x), or 1)}\)
\(\displaystyle \sum\limits_{i = 1}^{(1) + 1} {(i2^i )} = (1)2^{(1) + 2} + 2\)
\(\displaystyle (1*2^1 ) + (2*2^2 ) = 10\)
\(\displaystyle 2 + 8 = 10\)
\(\displaystyle 10 = 10\text{ [true for p(a)]}\)
\(\displaystyle \text{Step 2:}\)
\(\displaystyle \text{Suppose that p(k) holds for all k} \geqslant \text{a}\)
\(\displaystyle \sum\limits_{i = 1}^{k + 1} {(i2^i )} = k2^{k + 2} + 2\)
\(\displaystyle \text{I start from the left hand side using the difinition of}\sum {}\)
\(\displaystyle \sum\limits_{i = 1}^{(k + 1) + 1} {(i2^i )} = \sum\limits_{i = 1}^{k + 2} {(i2^i )} = (1)2^1 + (2)2^2 + (3)2^3 ...(k)2^k + (k + 1)2^{k + 1} + (k + 2)2^{k + 2}\)
\(\displaystyle \text{ = }\sum\limits_{i = 1}^{k + 1} {(i2^i )} \text{ + }(k + 2)2^{k + 2}\)
\(\displaystyle \text{Substitute for induction hypothesis}\)
\(\displaystyle \sum\limits_{i = 1}^{k + 1} {(i2^i )} = k2^{k + 2} + (2 + (k + 2)2^{k + 2} )\)
\(\displaystyle \text{Where do I go from here}...\text{Am I correct so far?}\)
