First slide

Arithmetic progression

Question

If the ratio of sums to n terms of two A.P.’s is $(5 n + 7) : (3 n + 2),$ then the ratio of their $17 t h$ terms is

Moderate

A

$172 : 99$
B

$172 : 101$
C

$175 : 99$
D

$175 : 101$

Solution

Let two A.P.'s be
$a, a + d, a + 2 d, a + 3 d, \dots$
and $A, A + D, A + 2 D, A + 3 D, \dots$
According to the given condition,
$\frac{\frac{n}{2} [2 a + (n - 1) d]}{\frac{n}{2} [2 A + (n - 1) D]} = \frac{5 n + 7}{3 n + 2} \Rightarrow \frac{a + \frac{n - 1}{2} d}{A + \frac{n - 1}{2} D} = \frac{5 n + 7}{3 n + 2}$
Putting $\frac{n - 1}{2} = 16$ or $n = 33,$ we get
$\frac{a + 16 d}{A + 16 D} = \frac{5 (33) + 7}{3 (33) + 2} = \frac{172}{101}$

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