Chapter 5: Sequences, Series and Progressions (Set-1)
In an AP, which value stays constant between consecutive terms?
A Common ratio
B Product constant
C Common difference
D Sum constant
In an arithmetic progression, the difference between any term and the previous term is always the same. This fixed value is called the common difference, denoted by d.
If an AP has first term a and common difference d, the nth term is
A a + nd
B a − nd
C (a+d)n
D a + (n−1)d
The first term is a. Each next term adds d. After (n−1) steps, total added is (n−1)d, so nth term becomes a + (n−1)d.
Which formula gives the sum of first n terms of an AP?
A n(a+l)/2
B a(r^n−1)/(r−1)
C n(a−l)/2
D a + (n−1)d
For an AP, sum of first n terms equals half of n times (first term + last term). Here last term is l. So Sₙ = n(a+l)/2.
If three numbers are in AP, the middle number equals
A Geometric mean
B Harmonic mean
C Arithmetic mean
D Weighted mean
In a three-term AP (x, y, z), the middle term is the average of extremes: y = (x+z)/2. That is the arithmetic mean of the first and third terms.
Condition for (a, b, c) to be in AP is
A b² = ac
B 2b = a + c
C a + b = c
D a/b = b/c
In an AP, the middle term equals the average of the other two. So b = (a+c)/2, which rearranges to 2b = a + c. This is the standard AP condition.
If d = 0 in an AP, the sequence is
A Increasing
B Decreasing
C Alternating
D Constant
Common difference d = 0 means each term equals the previous term. So all terms are the same and the AP becomes a constant sequence like a, a, a, a, ….
In an AP, if a₁ = 7 and d = 3, then a₄ equals
A 16
B 10
C 13
D 19
Use aₙ = a + (n−1)d. Here a₄ = 7 + (4−1)×3 = 7 + 9 = 16? Wait: 7+9=16, so correct is 16.
Correct value of a₄ for a₁=7, d=3 is
A 16
B 13
C 17
D 19
The 4th term is a₄ = a₁ + 3d. With a₁=7 and d=3, a₄ = 7 + 9 = 16. Each step adds 3: 7, 10, 13, 16.
If an AP has a = 5, d = −2, it is
A Always increasing
B Always constant
C Random terms
D Always decreasing
A negative common difference means each next term is smaller than the previous one. So the AP decreases steadily: 5, 3, 1, −1, … with constant drop of 2.
The last term l of an AP equals
A a + nd
B n(a+d)
C a + (n−1)d
D a − (n−1)d
The last term is the nth term of the AP. Using aₙ = a + (n−1)d, the last term l is exactly a + (n−1)d for n terms.
If a, b, c are in GP, then the middle term satisfies
A 2b = a+c
B b² = ac
C a+b = c
D ab = bc
In a three-term GP (a, b, c), the ratio b/a equals c/b. Multiplying gives b² = ac. This is the key condition to test a three-term GP.
In a GP, which value stays constant between consecutive terms?
A Common difference
B Constant sum
C Common ratio
D Constant average
A geometric progression has a fixed ratio r between consecutive terms: a₂/a₁ = a₃/a₂ = r. This constant multiplier is called the common ratio.
nth term of a GP with first term a and ratio r is
A a + (n−1)r
B arⁿ
C a(r−1)n
D arⁿ⁻¹
In GP, each step multiplies by r. Starting from a at n=1, after (n−1) multiplications the term becomes a·r^(n−1). That is the standard nth term formula.
Sum of first n terms of a GP (r ≠ 1) is
A a(1−rⁿ)/(1−r)
B n(a+l)/2
C a(1+rⁿ)/(1−r)
D a + (n−1)d
For GP with first term a and ratio r (not 1), the sum is Sₙ = a(1−rⁿ)/(1−r). It comes from subtracting rSₙ from Sₙ.
A GP with r = 1 becomes
A Alternating sequence
B Decreasing sequence
C Constant sequence
D Harmonic series
If r=1, every term equals a·1^(n−1)=a. So all terms are equal. Hence the GP becomes constant: a, a, a, … without growth or decay.
Sum to infinity of GP exists when
A r = 1
B |r| < 1
C r > 1
D r = −1
An infinite geometric series converges only when the ratio’s absolute value is less than 1. Then terms approach 0 and sum becomes S∞ = a/(1−r).
If a GP has a = 8 and r = 1/2, then second term is
A 8
B 16
C 2
D 4
In a GP, the second term is a·r. With a=8 and r=1/2, a₂ = 8×1/2 = 4. Each next term halves the previous term.
Geometric mean between 4 and 16 is
A 10
B 6
C 8
D 12
Geometric mean of two positive numbers x and y is √(xy). Here √(4×16) = √64 = 8. It is the middle term of a 3-term GP.
Arithmetic mean between 6 and 14 is
A 10
B 8
C 9
D 12
Arithmetic mean of two numbers is their average: (6+14)/2 = 20/2 = 10. It is the middle term when the three numbers form an arithmetic progression.
If AM between x and y is m, then
A m = xy
B m = √(x+y)
C m = x−y
D m = (x+y)/2
The arithmetic mean is defined as the sum divided by 2 for two numbers. So AM of x and y is (x+y)/2. This is used widely in AP problems.
If GM between x and y is g (x,y>0), then
A g = x+y
B g = x/y
C g = √(xy)
D g = (x+y)/2
Geometric mean of two positive numbers is the square root of their product. It matches the middle term property of a 3-term GP: x, √(xy), y.
Basic AM–GM inequality for x,y>0 is
A AM ≤ GM
B AM ≥ GM
C AM = GM always
D AM unrelated GM
For positive numbers, arithmetic mean is always at least the geometric mean. Equality happens only when x=y. This inequality is basic and helps in maxima-minima type questions.
AM equals GM when (x,y>0)
A x = y
B x > y
C x < y
D x + y = 0
AM–GM equality condition is that all involved positive numbers are equal. For two numbers, AM = GM happens only when x=y. Otherwise AM is strictly greater.
A sequence is best defined as
A Sum of terms
B Random set
C Product list
D Ordered list
A sequence is an ordered list of numbers written in a specific order, often following a rule. Examples include 1,2,3,… or 2,4,8,… where position matters.
A series is best defined as
A Ordered list
B Ratio of terms
C Sum of sequence
D Product of terms
A series is the sum of terms of a sequence, like a₁ + a₂ + a₃ + … . The value depends on partial sums, and it may converge or diverge.
Sigma notation ∑ aₖ represents
A Sum of terms
B Product of terms
C Difference of terms
D Ratio of terms
The symbol ∑ is used to represent summation. For example ∑ from k=1 to n aₖ means add a₁ + a₂ + … + aₙ. It is standard for series writing.
Partial sum Sₙ of a series means
A Sum to infinity
B Last two terms
C nth term only
D Sum of first n
Partial sum Sₙ is the sum of the first n terms of a series. Studying Sₙ as n increases helps decide whether a series converges or diverges.
If aₙ does NOT approach 0, the series ∑ aₙ
A Must converge
B Always equals 0
C Must diverge
D Always oscillates
A necessary condition for convergence of ∑aₙ is that aₙ → 0. If terms do not go to 0, partial sums cannot settle to a finite limit, so the series diverges.
Which is a telescoping type series form?
A 1 + 1 + 1
B 1/(n(n+1))
C 1/n
D rⁿ
1/(n(n+1)) can be split into 1/n − 1/(n+1). When summed, many terms cancel, leaving only a few boundary terms. This cancellation is telescoping.
Harmonic series is
A ∑ n
B ∑ rⁿ
C ∑ 1/n²
D ∑ 1/n
The harmonic series is 1 + 1/2 + 1/3 + … = ∑(1/n). It is a famous example of a series that diverges even though its terms go to 0.
Sum of first n natural numbers equals
A n²
B (n+1)/2
C n(n+1)/2
D n(n−1)/2
The sum 1+2+…+n equals n(n+1)/2. It can be derived by pairing first and last terms. This formula is often used in AP and series problems.
Sum of squares 1²+2²+…+n² equals
A n(n+1)(2n+1)/6
B n(n+1)/2
C n³
D (2n+1)/6
The standard formula for sum of first n squares is n(n+1)(2n+1)/6. It is useful in series questions and appears in many exam MCQs.
Sum of cubes 1³+2³+…+n³ equals
A n(n+1)(2n+1)/6
B n³(n+1)
C n²(n+1)
D [n(n+1)/2]²
The sum of cubes equals the square of the sum of first n natural numbers: (1+2+…+n)². So it becomes [n(n+1)/2]². This is a classic identity.
Fibonacci sequence starts as
A 1,3,5,7…
B 2,4,6,8…
C 0,1,1,2…
D 1,2,4,8…
Fibonacci sequence begins 0, 1, and each next term is the sum of previous two: 0,1,1,2,3,5,… It is a simple example of a recursive sequence.
A recursive sequence is defined using
A Only nth formula
B Previous terms
C Random values
D Only last term
In recursion, each term is described using earlier terms, like aₙ = aₙ₋₁ + 2. An initial term is needed, then the rule generates the full sequence step by step.
If aₙ = 3n + 2, the sequence is
A GP
B Harmonic
C Alternating
D AP
aₙ = 3n + 2 is linear in n, so consecutive differences are constant. Compute aₙ₊₁−aₙ = 3. Hence it forms an AP with common difference 3.
If aₙ = 5·2ⁿ, the sequence is
A AP
B Constant
C GP
D Telescoping
aₙ = 5·2ⁿ grows by multiplying by 2 each step. Ratio aₙ₊₁/aₙ = 2. So it is a geometric progression with common ratio 2.
If r is negative in a GP, terms are
A Alternate signs
B All positive
C All equal
D Always zero
A negative common ratio means each term multiplies by a negative number, flipping sign each step. For example 2, −4, 8, −16,… signs alternate between positive and negative.
If an AP has a=2, d=2, then first three terms are
A 2,3,4
B 2,4,6
C 2,6,10
D 2,2,2
In AP, add d each time. Starting at 2 with d=2 gives 2, then 4, then 6. This simple pattern helps verify understanding of common difference.
If a GP has a=3, r=3, then first three terms are
A 3,6,9
B 3,3,3
C 3,9,27
D 3,12,48
In GP, multiply by r each time. Starting with 3 and r=3: next is 9, next is 27. This shows repeated multiplication rather than repeated addition.
Arithmetic mean of 10 and 22 equals
A 14
B 18
C 20
D 16
AM = (10+22)/2 = 32/2 = 16. It represents the central value in an AP formed by 10, 16, 22. It is simple and widely used.
Geometric mean of 9 and 25 equals
A 12
B 18
C 15
D 10
GM = √(9×25) = √225 = 15. It is the middle term of the GP 9, 15, 25. Geometric mean requires numbers to be positive.
If AM of two numbers is 12 and numbers are equal, each is
A 12
B 6
C 24
D 10
If two numbers are equal, their arithmetic mean equals that same number. So if AM is 12, both numbers must be 12. This also matches the AM=GM equality case.
A bounded sequence means terms stay within
A One fixed point
B Infinite growth
C Random jumps
D Two fixed bounds
A sequence is bounded if all its terms lie between some lower bound and upper bound, like −5 ≤ aₙ ≤ 10. Boundedness is important in convergence discussions.
A monotonic increasing sequence means
A Always decreases
B Never decreases
C Random order
D Always constant
Monotonic increasing means aₙ₊₁ ≥ aₙ for all n. It can stay equal or rise, but it does not go down. Such sequences often help in convergence tests.
Alternating series typically has terms with
A Same sign
B Increasing values
C Changing sign
D Constant ratio 1
Alternating series has terms that switch sign, like 1 − 1/2 + 1/3 − 1/4 + … . The sign change is key in basic convergence ideas for alternating sums.
For |r|<1, sum to infinity of GP is
A a(1−r)
B a/(1+r)
C a(1−rⁿ)
D a/(1−r)
When |r|<1, the infinite GP converges and its sum is S∞ = a/(1−r). Because rⁿ approaches 0 as n becomes large, the finite sum formula simplifies.
If a GP has a=10 and r=0.1, sum to infinity is
A 10/0.9
B 10/1.1
C 10×0.9
D 10×1.1
Since |r|<1, S∞ = a/(1−r) = 10/(1−0.1) = 10/0.9. This equals 11.111… showing how decreasing GP can still have finite total sum.
If three numbers are in AP, their sum equals
A 2 × middle
B 3 × middle
C Middle minus 3
D Product of ends
For three-term AP (a−d, a, a+d), sum is (a−d)+a+(a+d)=3a. So the total equals three times the middle term. This is a useful shortcut.
If three numbers are in GP, their product equals
A Middle term
B Sum of ends
C Middle squared
D Difference ends
For three-term GP (a/r, a, ar), the product of extremes is (a/r)(ar)=a². Hence middle term squared equals product of first and third, giving the condition b² = ac.