JEE Advanced · 2015 · 24 May 2015 · Paper 2 · Sequences and Series

WhatsApp Email

Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6:11 and the seventh term lies in between 130 and 140, then the common difference of this A.P. is

Answer: 9

Solution

Let the A.P. have first term

a

and common difference

d

. Given

\frac{S _{7}}{S _{11}} = \frac{6}{11}

, we have

\frac{\frac{7}{2} ( 2 a + 6 d )}{\frac{11}{2} ( 2 a + 10 d )} = \frac{6}{11}

. Simplifying:

\frac{7 ( a + 3 d )}{11 ( a + 5 d )} = \frac{6}{11}

, which gives

7 a + 21 d = 6 a + 30 d

, so

a = 9 d

. The seventh term is

a + 6 d = 9 d + 6 d = 15 d

. Given

130 < 15 d < 140

, we get

8.67 < d < 9.33

. Since

d

is a natural number,

d = 9

JEE Advanced · 2015 · 24 May 2015 · Paper 2 · Integer answer

Practise more: all JEE Advanced PYQs on Sequences and Series →

More JEE Advanced questions on Sequences and Series

Let $R$ denote the set of all real numbers. Let $f : R \to R$ be a function such that $f (x) > 0$ for all $x \in R$ , and $f (x + y) = f (x) f (y)$ for all $x, y \in R$ .

Let the real numbers …
JEE Advanced 2025View solution →
Let $75...57$ denote the $(r + 2)$ digit number where the first and the last digits are 7 and the remaining r digits are 5. Consider the sum $S = 77 + 757 + 7557 + \dots + 75 \dots 57$ . If $S = \frac{75 \dots 57 + m}{n}$ , where m and n are natur …
JEE Advanced 2023View solution →
Let $l_{1}, l_{2}, \dots, l_{100}$ be consecutive terms of an arithmetic progression with common difference $d_{1}$ , and let $w_{1}, w_{2}, \dots, w_{100}$ be consecutive terms of another arithmetic progression with common difference $d_{2}$ , where $d_{1} d_{2} = 10$ . For each …
JEE Advanced 2022View solution →
Let $a_{1}, a_{2}, a_{3}, \dots$ be an arithmetic progression with $a_{1} = 7$ and common difference 8. Let $T_{1}, T_{2}, T_{3}, \dots$ be such that $T_{1} = 3$ and $T_{n + 1} - T_{n} = a_{n}$ for $n \geq 1$ . Then, which of the following is/are TRUE ?
JEE Advanced 2022View solution →
For positive integer n, define
$f (n) = n + \frac{16 + 5 n - 3 n ^{2}}{4 n + 3 n ^{2}} + \frac{32 + n - 3 n ^{2}}{8 n + 3 n ^{2}} + \frac{48 - 3 n - 3 n ^{2}}{12 n + 3 n ^{2}} + \dots + \frac{25 n - 7 n ^{2}}{7 n ^{2}} .$

Then, the value of $lim_{n \to \infty} f (n)$ is equal to
JEE Advanced 2022View solution →
Let
$M = {(x, y) \in R \times R : x^{2} + y^{2} \leq r^{2}},$

where $r > 0$ . Consider the geometric progression …
JEE Advanced 2021View solution →