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Math  /  Algebra

Question12.) For every natural number ' nn ', (n+1)2−n2(n+1)^{2}-n^{2} equals. (a) n−(n+1)n-(n+1) (b) (n+1)−n(n+1)-n (c) (n−1)+n(n-1)+n (d) (n+1)+n(n+1)+n

Studdy Solution

STEP 1

1. We are working with natural numbers n n , which are positive integers.
2. The expression (n+1)2−n2(n+1)^2 - n^2 can be simplified using algebraic techniques.
3. The goal is to find the simplest form of the expression and match it to one of the given options.

STEP 2

1. Expand the squares (n+1)2(n+1)^2 and n2n^2.
2. Subtract the expression for n2n^2 from the expression for (n+1)2(n+1)^2.
3. Simplify the resulting expression.
4. Compare the simplified expression to the given answer choices.

STEP 3

Expand (n+1)2(n+1)^2 using the distributive property.
(n+1)2=(n+1)(n+1)=n2+2n+1 (n+1)^2 = (n+1)(n+1) = n^2 + 2n + 1

STEP 4

Recognize that n2 n^2 is already in its simplest form.
n2=n2 n^2 = n^2

STEP 5

Subtract n2 n^2 from (n+1)2 (n+1)^2 .
(n+1)2−n2=(n2+2n+1)−n2 (n+1)^2 - n^2 = (n^2 + 2n + 1) - n^2

STEP 6

Simplify the resulting expression by combining like terms.
(n2+2n+1)−n2=2n+1 (n^2 + 2n + 1) - n^2 = 2n + 1

STEP 7

Compare the simplified expression 2n+1 2n + 1 to each of the given answer choices: - Option (a): n−(n+1)=n−n−1=−1 n - (n+1) = n - n - 1 = -1 - Option (b): (n+1)−n=n+1−n=1 (n+1) - n = n + 1 - n = 1 - Option (c): (n−1)+n=n−1+n=2n−1 (n-1) + n = n - 1 + n = 2n - 1 - Option (d): (n+1)+n=n+1+n=2n+1 (n+1) + n = n + 1 + n = 2n + 1
The correct match is: (d) (n+1)+n (n+1) + n
The solution to the problem is option (d): (n+1)+n (n+1) + n .

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