Chapter 12: Q. 6 (page 830)

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n :
$1 + 4 + 7 + . . . + (3 n - 2) = \frac{1}{2} n (3 n - 1)$

Short Answer

Expert verified

The statement is true for all $n \in N$

Step by step solution

. Given Information

We are given a statement :

$1 + 4 + 7 + . . . + (3 n - 2) = \frac{1}{2} n (3 n - 1)$

. Checking for n=1

$P (1) = 1 (L . H . S)$

$P (1) = \frac{1 (3 - 1)}{2} = 1 (R . H . S)$

Therefore the statement is true for n=1

. Checking for n=k

Let P(n) is true for n=k so we need to prove that P(k+1) is true : $P (k) = 1 + 4 + 7 + . . . + (3 k - 2) = \frac{1}{2} k (3 k - 1)$ -equation(i)

So, the next term will be :

$1 + 4 + 7 + . . . + (3 k - 2) + (3 k + 1) = \frac{(k + 1) (3 k + 2)}{2}$

Now we need to modify the L.H.S so that it is equal to the R.H.S, we can substitute the value of equation one in this :

$\frac{k (3 k - 1) + 2 (3 k + 1)}{2} = \frac{3 k^{2} - k + 6 k + 2}{2}$

$\frac{3 k^{2} + 5 k + 2}{2} = \frac{(k + 1) (3 k + 2)}{2}$

So, R.H.S = L.H.S which means P(n) is true for n=k+1

Hence , the statement is true for all $n \in N$

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Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n :
$1 + 4 + 7 + . . . + (3 n - 2) = \frac{1}{2} n (3 n - 1)$

Short Answer

Step by step solution

. Given Information

. Checking for n=1

. Checking for n=k

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Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n :1+4+7+...+(3n-2)=12n(3n-1)

Short Answer

Step by step solution

. Given Information

. Checking for n=1

. Checking for n=k

One App. One Place for Learning.

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Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n :
$1 + 4 + 7 + . . . + (3 n - 2) = \frac{1}{2} n (3 n - 1)$