Assertion (A): 1 + (2 + 3) + (4 + 5 + 6) + ... n terms = sum_{r=1}^{n} r(r^2 + 1)/2, n in N Reason (R): n^{th} term of sequence 1 + 2 + 4 + 7 + ... is (n^2 - n + 2)/2, n in N and sum to n terms of an arithmetic progression {T_n: a + (n-1)d} is n/2[2a + (n-1)d]

The correct answer is Option 1. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q3 – AP-EAMCET 2026

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Home Maths Sequences and SeriesQuestion #387

Assertion (A):

1 + (2 + 3) + (4 + 5 + 6) + \dots n terms = \sum_{r = 1}^{n} \frac{r (r^{2} + 1)}{2}, n \in N

Reason (R):

n^{th}

term of sequence

1 + 2 + 4 + 7 + \dots

\frac{n^{2} - n + 2}{2}, n \in N

and sum to n terms of an arithmetic progression

{T_{n} : a + (n - 1) d}

\frac{n}{2} [2 a + (n - 1) d]

Clue 1 of 3: Understand the structure of Assertion (A)

Assertion (A) describes a series where each term is itself the sum of an arithmetic progression of natural numbers.

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Step 1: Evaluate Assertion (A)✦ Active

Let the given series be $S = 1 + (2 + 3) + (4 + 5 + 6) + \dots n terms$ . Let $T_{k}$ denote the $k$ -th term of this series. The $k$ -th term $T_{k}$ is the sum of $k$ consecutive natural numbers. The natural numbers forming $T_{k}$ start after all numbers from previous terms $T_{1}, T_{2}, \dots, T_{k - 1}$ have been used. The total count of numbers used before the $k$ -th term begins is $1 + 2 + \dots + (k - 1) = \frac{(k - 1) k}{2}$ . Thus, the first number in the $k$ -th term $T_{k}$ is $a_{k} = \frac{(k - 1) k}{2} + 1$ . The term $T_{k}$ is an arithmetic progression of $k$ terms with first term $a_{k}$ and common difference $d = 1$ . Using the sum of an A.P. formula $S_{m} = \frac{m}{2} [2 a + (m - 1) d]$ with $m = k, a = a_{k}, d = 1$ :

T_{k} = \frac{k}{2} [2 (\frac{(k - 1) k}{2} + 1) + (k - 1) (1)]

T_{k} = \frac{k}{2} [k (k - 1) + 2 + k - 1]

T_{k} = \frac{k}{2} [k^{2} - k + k + 1]

T_{k} = \frac{k (k^{2} + 1)}{2}

The assertion states that the sum to $n$ terms is $\sum_{r = 1}^{n} \frac{r (r^{2} + 1)}{2}$ . This matches the formula for the $k$ -th term we derived. Therefore, Assertion (A) is **True**.

💡 Teacher's Secret Hint

Remember that the 'n terms' in the series refers to 'n groups', not 'n individual numbers'. Each group is itself an arithmetic progression.

Step 2: Evaluate Reason (R)○ Expand

Reason (R) consists of two parts. Let's evaluate each:

Part 1: The $n^{th}$ term of the sequence $1, 2, 4, 7, \dots$ is $\frac{n^{2} - n + 2}{2}$ . Let the sequence be $a_{n}$ . The terms are $a_{1} = 1, a_{2} = 2, a_{3} = 4, a_{4} = 7$ . The first differences are $a_{2} - a_{1} = 1, a_{3} - a_{2} = 2, a_{4} - a_{3} = 3$ . Since the first differences form an arithmetic progression ( $1, 2, 3, \dots$ ), the general term $a_{n}$ must be a quadratic polynomial of the form $a_{n} = A n^{2} + B n + C$ . By substituting the first few terms:

n = 1 : A + B + C = 1

n = 2 : 4 A + 2 B + C = 2

n = 3 : 9 A + 3 B + C = 4

Subtracting equations (2)-(1) gives $3 A + B = 1$ . Subtracting (3)-(2) gives $5 A + B = 2$ . Subtracting these new equations gives $(5 A + B) - (3 A + B) = 2 - 1 ⟹ 2 A = 1 ⟹ A = \frac{1}{2}$ . Substituting $A = \frac{1}{2}$ into $3 A + B = 1$ gives $B = 1 - 3 (\frac{1}{2}) = 1 - \frac{3}{2} = - \frac{1}{2}$ . Substituting $A = \frac{1}{2}$ and $B = - \frac{1}{2}$ into $A + B + C = 1$ gives $\frac{1}{2} - \frac{1}{2} + C = 1 ⟹ C = 1$ . Thus, $a_{n} = \frac{1}{2} n^{2} - \frac{1}{2} n + 1 = \frac{n^{2} - n + 2}{2}$ . This statement is **True**.

Part 2: Sum to $n$ terms of an arithmetic progression ${T_{n} : a + (n - 1) d}$ is $\frac{n}{2} [2 a + (n - 1) d]$ . This is the standard formula for the sum of $n$ terms of an arithmetic progression. This statement is **True**.

Since both parts of Reason (R) are true, Reason (R) is **True**.

💡 Teacher's Secret Hint

When finding the general term of a sequence whose first differences form an AP, the general term is a quadratic polynomial. If the second differences form an AP, it's a cubic, and so on.

Step 3: Determine if (R) is the correct explanation for (A)○ Expand

Let's check if Reason (R) explains Assertion (A). We found that the $k$ -th term of the series in Assertion (A) starts with a specific natural number. Let's compare these starting numbers with the sequence given in Reason (R):

∙

For

k = 1

T_{1} = 1

. The first term of the sequence in R is

\frac{1^{2} - 1 + 2}{2} = 1

∙

For

k = 2

T_{2} = 2 + 3

. The first number is

2

. The second term of the sequence in R is

\frac{2^{2} - 2 + 2}{2} = 2

∙

For

k = 3

T_{3} = 4 + 5 + 6

. The first number is

4

. The third term of the sequence in R is

\frac{3^{2} - 3 + 2}{2} = 4

∙

For

k = 4

T_{4} = 7 + 8 + 9 + 10

. The first number is

7

. The fourth term of the sequence in R is

\frac{4^{2} - 4 + 2}{2} = 7

This confirms that the first part of Reason (R) (the $n^{th}$ term of the sequence $1, 2, 4, 7, \dots$ ) gives the starting term for the $n$ -th group in Assertion (A). To calculate the sum of each group (which is $T_{k}$ ), we use the formula for the sum of an arithmetic progression, which is the second part of Reason (R). Since both parts of Reason (R) are directly used and are essential steps in deriving the terms and thus the sum in Assertion (A), Reason (R) is the correct explanation for Assertion (A).

💡 Teacher's Secret Hint

The wording 'correct explanation' implies that Reason (R) should provide the underlying principles or a direct derivation for Assertion (A). In this case, both elements of (R) are directly applied in the calculation of (A).

Step 4: Conclusion○ Expand

Based on the analysis: 1. Assertion (A) is **True**. 2. Reason (R) is **True**. 3. Reason (R) is the correct explanation for Assertion (A).

This corresponds to Option 1.

💡 Teacher's Secret Hint

Always verify both the truth value of A and R, and then their relationship. Don't assume the relationship based solely on A and R being true.

✦ STEM Console utilizes AI models to generate step-by-step explanations and math clues. AI can make mistakes.

Maths Question 3 – AP-EAMCET 2026

Clue 1 of 3: Understand the structure of Assertion (A)

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