First slide

Arithmetic progression

Question

The sum of first 24 terms of the A.P, $a_{1}, a_{2}, a_{3}, ....$ if it is known that $a_{1} + a_{5} + a_{10} + a_{15} + a_{20} + a_{24} = 225$

Moderate

A

700
B

800
C

900
D

600

Solution

We known that in an A.P the sum of the terms
equidistant from the beginning and end is always same and is equal to
the sum of first and last term, ie. $a_{1} + a_{n} = a_{2} + a_{n - 1} = a_{3} + a_{n - 2} = .....$
So if an A.P. consists of 24 terms, then
$\begin{array}{l} a_{1} + a_{24} = a_{3} + a_{20} = a_{10} + a_{15} \\ a_{1} + a_{5} + a_{10} + a_{15} + a_{20} + a_{24} = 225 \\ \Rightarrow a_{1} + a_{24} = \frac{225}{3} = 75 \end{array}$
$\begin{array}{l} ∴ S_{24} = \frac{24}{2} (a_{1} + a_{24}) [u \sin g S_{n} = \frac{n}{2} (a_{1} + a_{n})] \\ = 12 (75) \\ = 900 (using (i)) \end{array}$

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