Basic Definitions — JEE Main Mathematics MCQs with Solutions
Free JEE Main Mathematics Basic Definitions MCQs with step-by-step solutions (21 questions). Part of Sequences and Series. Practise online on Prepizo — no login needed.
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Questions with solutions
Q1 — Basic Definitions · easy · theory
A sequence is best described as a function whose domain is:
A. the empty set
B. the set of rational numbers
C. the set of real numbers
D. the set of natural numbers ✓ Correct
Solution: A sequence is a function $f:\mathbb{N} \to \mathbb{R}$; its values $f(1), f(2), \dots$ are the terms.
Q2 — Basic Definitions · easy · theory
A series is obtained from a sequence by:
A. multiplying its terms
B. reversing its terms
C. adding its terms ✓ Correct
D. taking reciprocals of its terms
Solution: If $a_1, a_2, \dots$ is a sequence, the expression $a_1 + a_2 + \dots$ is the corresponding series.
Q3 — Basic Definitions · easy · numerical
If the general term of a sequence is $T_n = 2n + 3$, then $T_5$ equals:
A. $11$
B. $10$
C. $15$
D. $13$ ✓ Correct
Solution: $T_5 = 2(5) + 3 = 13$.
Q4 — Basic Definitions · easy · numerical
For the sequence with $T_n = n^2 - n$, the value of $T_4$ is:
A. $8$
B. $16$
C. $12$ ✓ Correct
D. $20$
Solution: $T_4 = 4^2 - 4 = 16 - 4 = 12$.
Q5 — Basic Definitions · medium · numerical
The general term of the sequence $3, 8, 15, 24, \dots$ is:
A. $2n^2 - 1$
B. $n^2 + 1$
C. $n^2 + 2n$ ✓ Correct
D. $3n$
Solution: The terms are $n(n+2)$: for $n=1,2,3,4$ we get $3, 8, 15, 24$.
Q6 — Basic Definitions · medium · numerical
The general term of the sequence $2, 5, 10, 17, 26, \dots$ is:
A. $n^2 + 1$ ✓ Correct
B. $2n + 1$
C. $n^2 + n$
D. $3n - 1$
Solution: The terms are $n^2 + 1$: $2, 5, 10, 17, 26$ for $n = 1,2,3,4,5$.
Q7 — Basic Definitions · medium · numerical
If the sum of the first $n$ terms of a sequence is $S_n = n^2 + 2n$, then its $n$-th term $T_n$ is:
A. $2n + 1$ ✓ Correct
B. $n^2$
C. $n + 2$
D. $2n - 1$
Solution: $T_n = S_n - S_{n-1} = (n^2+2n) - ((n-1)^2 + 2(n-1)) = 2n + 1$.
Q8 — Basic Definitions · medium · numerical
If $S_n = 3n^2 - 2n$ is the sum of $n$ terms, then $T_n$ equals:
A. $6n - 5$ ✓ Correct
B. $6n - 2$
C. $6n + 5$
D. $3n - 2$
Solution: $T_n = S_n - S_{n-1} = (3n^2-2n) - (3(n-1)^2 - 2(n-1)) = 6n - 5$.
Q9 — Basic Definitions · medium · numerical
A sequence is defined by $a_1 = 1$ and $a_{n+1} = a_n + 2n$. Then $a_4$ equals:
A. $13$ ✓ Correct
B. $9$
C. $11$
D. $15$
Solution: $a_2 = 1+2 = 3,\ a_3 = 3+4 = 7,\ a_4 = 7+6 = 13$.
Q10 — Basic Definitions · medium · numerical
The number of terms in the finite sequence $5, 8, 11, \dots, 50$ is:
A. $16$ ✓ Correct
B. $17$
C. $15$
D. $14$
Solution: This is an A.P. with $a=5, d=3$: $5 + 3(n-1) = 50 \Rightarrow n = 16$.
Q11 — Basic Definitions · medium · theory
Which of the following represents an infinite series?
A. $2 + 4 + 6 + \cdots + 50$
B. $1 + 2 + 3 + \cdots + 100$
C. $1 + \tfrac12 + \tfrac14 + \tfrac18 + \cdots$ ✓ Correct
D. $5 + 10 + 15 + 20$
Solution: An infinite series has unending terms; $1 + \tfrac12 + \tfrac14 + \cdots$ continues without end.
Q12 — Basic Definitions · easy · numerical
The general term of the sequence $\tfrac12, \tfrac23, \tfrac34, \tfrac45, \dots$ is:
A. $\dfrac{n}{n-1}$
B. $\dfrac{n}{n+1}$ ✓ Correct
C. $\dfrac{n+1}{n}$
D. $\dfrac{1}{n+1}$
Solution: Each term is $\tfrac{n}{n+1}$ for $n = 1, 2, 3, \dots$
Q13 — Basic Definitions · medium · numerical
If $T_n = (-1)^n \, n$, the sum of the first four terms of the sequence is:
A. $10$
B. $0$
C. $2$ ✓ Correct
D. $-2$
Solution: $T_1 + T_2 + T_3 + T_4 = -1 + 2 - 3 + 4 = 2$.
Q14 — Basic Definitions · medium · numerical
Which term of the sequence $T_n = 4n - 3$ is equal to $45$?
A. $13$-th
B. $10$-th
C. $11$-th
D. $12$-th ✓ Correct
Solution: $4n - 3 = 45 \Rightarrow 4n = 48 \Rightarrow n = 12$.
Q15 — Basic Definitions · hard · numerical
If the sum of $n$ terms of a sequence is $S_n = 2^n - 1$, then its $10$-th term is:
A. $1023$
B. $256$
C. $1024$
D. $512$ ✓ Correct
Solution: $T_{10} = S_{10} - S_9 = (2^{10}-1) - (2^9 - 1) = 1024 - 512 = 512 = 2^9$.
Q16 — Basic Definitions · medium · numerical
For the sequence $a_1 = 2$ and $a_n = 3a_{n-1}$, the value of $a_4$ is:
A. $18$
B. $162$
C. $48$
D. $54$ ✓ Correct
Solution: $a_2 = 6,\ a_3 = 18,\ a_4 = 54$ (equivalently $2\cdot 3^3 = 54$).
Q17 — Basic Definitions · hard · numerical
The general term of the sequence $5, 11, 19, 29, \dots$ (successive differences $6, 8, 10, \dots$) is:
A. $3n^2 - 1$
B. $n^2 + 3n + 1$ ✓ Correct
C. $2n^2 + 3$
D. $n^2 + 2n + 2$
Solution: Differences form an A.P., so $T_n$ is quadratic: $T_n = n^2 + 3n + 1$ fits $5, 11, 19, 29$.
Q18 — Basic Definitions · medium · numerical
The $5$-th term of the sequence whose $n$-th term is $T_n = \dfrac{n(n+1)}{2}$ (triangular numbers) is:
A. $21$
B. $10$
C. $15$ ✓ Correct
D. $12$
Solution: $T_5 = \tfrac{5 \cdot 6}{2} = 15$.
Q19 — Basic Definitions · medium · theory
For a sequence, the statement "$S_n - S_{n-1} = T_n$" is valid for:
A. no value of $n$
B. all $n \ge 1$ automatically
C. only even $n$
D. $n \ge 2$, with $T_1 = S_1$ ✓ Correct
Solution: The difference formula gives $T_n$ for $n \ge 2$; the first term is found separately as $T_1 = S_1$.
Q20 — Basic Definitions · hard · numerical
If $T_n = \dfrac{1}{n(n+1)}$, then $T_1 + T_2 + T_3$ equals:
A. $\dfrac{2}{3}$
B. $\dfrac{11}{12}$
C. $\dfrac{3}{4}$ ✓ Correct
D. $\dfrac{1}{4}$
Solution: $\tfrac12 + \tfrac16 + \tfrac{1}{12} = \tfrac{6+2+1}{12} = \tfrac{9}{12} = \tfrac34$.
Q21 — Basic Definitions · hard · numerical
The $n$-th term of the sequence $2, 5, 10, 17, \dots$ increased by the $n$-th term of $1, 2, 3, 4, \dots$ gives, for $n = 4$:
A. $21$ ✓ Correct
B. $18$
C. $20$
D. $26$
Solution: First sequence $T_n = n^2 + 1 \Rightarrow T_4 = 17$; adding $4$ gives $21$.