How many terms of the AP 21, 18, 15, ... must be added to get the sum 0?
To obtain a sum of zero in the AP 21, 18, 15, ..., the first 15 terms must be added, using the arithmetic progression sum formula and solving for n = 15.
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What Is This AP and Why Does Its Sum Reach Zero?
The sequence 21, 18, 15, … is a decreasing arithmetic progression (AP). Each term is 3 less than the previous one, so the common difference d = −3. The terms start positive but steadily cross zero (the 8th term equals zero) and then become negative. Because positive early terms and negative later terms exist in the same sequence, their contributions to the running total will eventually cancel each other out — producing a sum of exactly zero.This type of problem appears in NCERT Class 10 Maths, Chapter 5 (Arithmetic Progressions) and is a favourite in school board exams, competitive scholarship tests, and JEE foundation papers. It tests two skills simultaneously: correctly applying the sum formula and solving the resulting quadratic (or linear) equation without discarding valid roots carelessly.
How many terms of the AP 21, 18, 15, ... must be added to get the sum 0?
The given AP is 21, 18, 15, ... .
Here, a = 21 and d = 18 − 21 = −3
Let the required number of terms be n. Then,
$S_{n}=0$
$\Rightarrow \frac{n}{2}[2 \times 21+(n-1) \times(-3)]=0 \quad\left\{S_{n}=\frac{n}{2}[2 a+(n-1) d]\right\}$
$\Rightarrow \frac{n}{2}(42-3 n+3)=0$
$\Rightarrow n(45-3 n)=0$
$\Rightarrow n=0$ or $45-3 n=0$
$\Rightarrow n=0$ or $n=15$
∴ n = 15 (Number of terms cannot be zero)
Hence, the required number of terms is 15.
Frequently Asked Questions
Find answers to common questions.
How many terms of the AP 21, 18, 15 must be added to get the sum 0?
15 terms must be added. Setting the sum formula Sₙ = n/2 [2(21) + (n−1)(−3)] equal to zero gives n(45 − 3n) = 0, yielding n = 0 or n = 15. Since n = 0 is invalid, the answer is n = 15. You can verify this by noting that S₁₅ = 15/2 × [42 + 14 × (−3)] = 15/2 × 0 = 0.
What is the common difference of the AP 21, 18, 15?
The common difference is d = −3. Each term decreases by 3: 18 − 21 = −3, 15 − 18 = −3. Because d is negative and the first term is positive, this is a decreasing AP whose terms will eventually become zero and then negative.
What is the formula used to find the sum of n terms of an AP?
The formula is Sₙ = n/2 [2a + (n−1)d], where a is the first term, d is the common difference, and n is the number of terms. An equivalent form is Sₙ = n/2 (a + l), where l is the last term — but this form requires knowing l in advance, so the first version is more broadly useful.
Which term of the AP 21, 18, 15 is zero?
The 8th term equals zero. Using aₙ = a + (n−1)d: a₈ = 21 + (8−1)(−3) = 21 − 21 = 0. This zero term sits at the centre of symmetry and is why the positive and negative portions of the sequence cancel out perfectly when 15 terms are summed.
Can the sum of an AP be zero for more than one value of n?
Yes, it can. If the AP has both positive and negative terms and a non-zero middle section, the quadratic Sₙ = 0 may produce two positive integer values of n. In such cases, both are valid answers — the sum crosses zero twice. Always check both roots of the equation rather than accepting just one.