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Finding n when sum is known

Finding n when sum is known means using the sum formula to determine how many terms of an AP add up to a given total.

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Student-friendly explanation

Sometimes the exam gives the total sum and asks for the number of terms. In such questions, students must set up the sum formula, substitute the known values, and solve the resulting equation for n. This often creates a quadratic equation, so neat algebra is important.

How to write this in exams

  1. 1

    Start with the exact idea

    Finding n when sum is known means using the sum formula to determine how many terms of an AP add up to a given total.

  2. 2

    Then show how to use it

    1. Write the sum formula. 2. Replace a, d, and S_n with the given values. 3. Simplify into an equation in n. 4. Solve for the positive whole number value of n.

  3. 3

    Add one concrete example

    If the sum of an AP is 30 and the terms are 2, 4, 6, 8, ..., then n = 5 because 2 + 4 + 6 + 8 + 10 = 30.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to stop after forming the equation and not solve it completely for n.

Definition

Finding n when sum is known means using the sum formula to determine how many terms of an AP add up to a given total.

Example

If the sum of an AP is 30 and the terms are 2, 4, 6, 8, ..., then n = 5 because 2 + 4 + 6 + 8 + 10 = 30.

Rule to remember

Use S_n = n/2 [2a + (n - 1)d] and solve for n when S_n is known.

Memory hook

When the total is known, n is the hidden count you must uncover.

Examples and method

Worked example

For the AP 3, 6, 9, 12, ... and S_n = 45, we get n/2[6 + 3(n - 1)] = 45. Solving gives n = 5.

Method to apply

1. Write the sum formula. 2. Replace a, d, and S_n with the given values. 3. Simplify into an equation in n. 4. Solve for the positive whole number value of n.

Diagram support

A table of term number and value can help verify the number of terms after solving the equation.

How CBSE asks it

CBSE may give the sum and ask for the number of terms, especially in money, seating, or counting problems.

Avoid common mistakes

Common confusion

Students may use the nth-term formula instead of the sum formula when the question is about total sum.

Common wrong answer

A common wrong answer is to stop after forming the equation and not solve it completely for n.

Exam tip

If the phrase says sum, total, or altogether, start with the sum formula.

Quick check

What should you do first when the sum of an AP is given and n is unknown?

You should write the sum formula and substitute the known first term, common difference, and total sum. Then solve the resulting equation for n.

Answer writing and exam use

1-mark answer

Finding n when sum is known means using the sum formula to determine how many terms of an AP add up to a given total.

2-mark answer

Finding n when sum is known means using the sum formula to determine how many terms of an AP add up to a given total. Use S_n = n/2 [2a + (n - 1)d] and solve for n when S_n is known. If the sum of an AP is 30 and the terms are 2, 4, 6, 8, ..., then n = 5 because 2 + 4 + 6 + 8 + 10 = 30.

3-mark answer

Sometimes the exam gives the total sum and asks for the number of terms. In such questions, students must set up the sum formula, substitute the known values, and solve the resulting equation for n. This often creates a quadratic equation, so neat algebra is important. Use S_n = n/2 [2a + (n - 1)d] and solve for n when S_n is known. For the AP 3, 6, 9, 12, ... and S_n = 45, we get n/2[6 + 3(n - 1)] = 45. Solving gives n = 5. CBSE may give the sum and ask for the number of terms, especially in money, seating, or counting problems. A common wrong answer is to stop after forming the equation and not solve it completely for n.
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