If the sum of n terms of an A.P. is n² + 2n, determine the AP.
Understand the Problem
The question is asking us to determine the arithmetic progression (AP) given the sum of its first n terms expressed as n² + 2n. We will use the formula for the sum of an AP to find the first term and the common difference.
Answer
The arithmetic progression is given by \( a_n = 2n + 1 \).
Answer for screen readers
The arithmetic progression is given by the formula: $$ a_n = 2n + 1 $$
Steps to Solve
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Identify the sum formula The sum of the first ( n ) terms of an arithmetic progression (AP) is given by the formula: $$ S_n = \frac{n}{2} (2a + (n - 1)d) $$ where ( a ) is the first term, ( d ) is the common difference, and ( S_n ) is the sum of the first ( n ) terms.
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Set the equation According to the problem, we know: $$ S_n = n^2 + 2n $$ We will equate this to the sum formula: $$ \frac{n}{2} (2a + (n - 1)d) = n^2 + 2n $$
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Clear the fraction Multiply both sides of the equation by 2 to eliminate the fraction: $$ n(2a + (n - 1)d) = 2(n^2 + 2n) $$
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Expand and simplify Expanding the right-hand side gives: $$ n(2a + (n - 1)d) = 2n^2 + 4n $$ Now divide both sides by ( n ) (assuming ( n \neq 0 )): $$ 2a + (n - 1)d = 2n + 4 $$
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Extract terms related to ( n ) Now, let's rearrange to separate the constant and ( n ) terms:
- The left side has ( 2a ) (a constant) and ( (n - 1)d ) (which is ( nd - d )).
- The right side shows ( 2n + 4 ). From this, we can observe that the coefficient of ( n ) gives ( d = 2 ) and the constant term ( 2a - d = 4 ).
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Solve for ( a ) Substituting ( d = 2 ) into the equation: $$ 2a - 2 = 4 $$ Solving for ( a ), we get: $$ 2a = 6 \implies a = 3 $$
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Write the AP Now, with ( a = 3 ) and ( d = 2 ), the arithmetic progression can be expressed as: $$ a_n = a + (n - 1)d = 3 + (n - 1)2 = 2n + 1 $$
The arithmetic progression is given by the formula: $$ a_n = 2n + 1 $$
More Information
The solution shows that the arithmetic progression starts with the first term of ( 3 ) and has a common difference of ( 2 ). This means the first few terms of the AP are ( 3, 5, 7, 9, \ldots ).
Tips
- Forgetting to clear the fraction properly can lead to incorrect terms.
- Confusing the constant term with the variable term ( d ) and making incorrect substitutions.
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