First slide

Arithmetic progression

Question

Let $a_{1}, a_{2}, a_{3} . . . .$ be terms of an A.P. If $\frac{a_{1} + a_{2} + \dots + a_{p}}{a_{1} + a_{2} + \dots + a_{q}} = \frac{p^{2}}{q^{2}}, p \neq q$ then $\frac{a_{6}}{a_{21}}$

Moderate

A

$\frac{11}{41}$
B

$\frac{41}{11}$
C

$\frac{7}{2}$
D

$\frac{2}{7}$

Solution

$\frac{\frac{p}{2} [2 a + (p - 1) d]}{\frac{q}{2} [2 a + (q - 1) d]} = \frac{p^{2}}{q^{2}} \Rightarrow \frac{a + \frac{p - 1}{2} d}{a + \frac{q - 1}{2} d} = \frac{p}{q}$
Now, put $p = 11$ and $q = 41 .$

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