Let $a_1, a_2, a_3, \ldots$ be a sequence of real numbers. Let $s_n=a_1+a_2+\cdots+a_n$. If $2 s_n=$...

To determine the progression of the sequence $a_1, a_2, a_3, \ldots$, we analyze the given condition for the sequence's sum:$ s_n = a_1 + a_2 + a_3 + \cdots + a_n = \frac{n}{2}(c + a_n) $This expression resembles the sum formula for an arithmetic progression (AP):$ s_n = \frac{n}{2}(a_1 + a_n) $To validate if our sequence is an AP, let's assume it is. If so, the sum formula for the first $n$ terms of an AP, where $a_1$ is the first term and $d$ is the common difference, would be:$ s_n = \frac{n}{2}(2a_1 + (n-1)d) $$ a_n = a_1 + (n-1)d $Substituting these into the given equation:$ 2\left(\frac{n}{2}(2a_1 + (n-1)d)\right) = n(c + a_1 + (n-1)d) $Simplifying both sides gives:$ 2a_1 + (n-1)d = c + a_1 + (n-1)d $This reduces further to:$ 2a_1 = a_1 + c $$ c = a_1 $Since $c = a_1$, this confirms our sequence $a_1, a_2, a_3, \ldots$ is indeed an arithmetic progression (AP), given that this assumption led to a consistent and valid result. Therefore, the assumption that $a_1, a_2, a_3, \ldots$ is an arithmetic progression is correct.

Let $S_n$ denote the sum of the first $n$ terms of a sequence $a_1, a_2, a_3, \ldots$. If $S_{n+3}-S...

We are told:$ S_{n+3}-S_n = 13n + 7, \quad \forall n. $But $S_n = a_1 + a_2 + ... + a_n$.So$ S_{n+3} - S_n = (a_{n+1} + a_{n+2} + a_{n+3}) = 13n + 7. \tag{1} $Step 1:We can write equations for different $n$.For $n, n+1, n+2,$ etc.At $n$:$ a_{n+1} + a_{n+2} + a_{n+3} = 13n + 7. $At $n+1$:$ a_{n+2} + a_{n+3} + a_{n+4} = 13(n+1) + 7 = 13n + 20. $Now subtract the two:$ (a_{n+2}+a_{n+3}+a_{n+4}) - (a_{n+1}+a_{n+2}+a_{n+3}) = (13n + 20) - (13n + 7). $Simplify:$ a_{n+4} - a_{n+1} = 13. $Step 2:So we have the recurrence:$ a_{n+4} = a_{n+1} + 13. $This means the sequence increases by 13 every 3 steps (since $a_{n+4}-a_{n+1}=13$).So for $a_{13} - a_{10}$:These indices differ by $3$:$ a_{13} - a_{10} = 13. $✅Final Answer:$a_{13} - a_{10} = 13.$Answer: Option A (13)

Sequences And Series - IAT-IISER Maths PYQs

Q: A lab reports that the global average temperature in the year 2020 was $14.9^{\circ} \mathrm{C}$ and...

We treat the 1 % increase per year as exponential growth over 15 years (from 2020 to 2035):Initial temperature: $T_0 = 14.9^\circ\!C$Annual growth factor: $1 + r = 1 + 0.01 = 1.01$Number of years: $n = 2035 - 2020 = 15$Future temperature:$$ T_{2035} = T_0 \times (1.01)^{15} = 14.9 \times (1.01)^{15} \approx 17.298^\circ\!C $$That matches Option B.

01

PYQ 2022

easy

maths ID: iat-iise

A lab reports that the global average temperature in the year 2020 was and predicts that the global average temperature will increase at the rate of per year. What will be the global average temperature in the year 2035?

1

2

3

4

02

PYQ 2024

medium

maths ID: iat-iise

Let be a sequence of real numbers. Let . If for some real number. for some real number and for all, then which one of the following statements is Correct?

1

is an Arithmetic Progression.

2

is an Arithmetic Progression.

3

is a Geometric Progression.

4

, ...is a Geometric Progression.

03

PYQ 2025

easy

maths ID: iat-iise

Let denote the sum of the first terms of a sequence . If for all, what is the value of ?

1

13

2

137

3

46

4

12

Sequences And Series

High-Yield Trend

Questions
3 MCQs

Official Solution

Official Solution

Official Solution

Sequences And Series

High-Yield Trend

Questions 3 MCQs

Official Solution

Official Solution

Official Solution

Questions
3 MCQs