Name: How to Identify Missing Number in the Sequence?
Uploaded: 2019-07-19
Duration: 14 min 44 s
Description: Let's observe the pattern in the given number of terms that follow some sequence and find out the missing term. Use your logic and try to solve it.

Question

The ratio of the sum of $n$ terms of two A.P.s is $(3 n + 4) : (5 n + 6)$ . Find the ratio of their $7 t h$ term.

Accepted Answer

There are two A.P.s with different first term and common difference

For the first A.P.

Let first term be $a$

Common difference be $d$

Sum of n terms is $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$

and $n t h$ term of an AP is $t_{n} = a + (n - 1) d$

For the second A.P.

Let first term be $A$

Common difference be $D$

Sum of n terms is $S_{n} = \frac{n}{2} [2 A + (n - 1) D]$

and $n t h$ term of an AP is $t_{n} = A + (n - 1) D$

Now need to find ratio of their $7 t h$ term

given that the ratio of the sum of $n$ terms of two A.P.s is $(3 n + 4) : (5 n + 6)$

$\Rightarrow \frac{\frac{n}{2} [2 a + (n - 1) d]}{\frac{n}{2} [2 A + (n - 1) D]} = \frac{3 n + 4}{5 n + 6}$

$\Rightarrow \frac{[2 a + (n - 1) d]}{[2 A + (n - 1) D]} = \frac{3 n + 4}{5 n + 6} \Rightarrow \frac{2 [a + (\frac{n - 1}{2}) d]}{2 [A + (\frac{n - 1}{2}) D]} = \frac{3 n + 4}{5 n + 6} \Rightarrow \frac{[a + (\frac{n - 1}{2}) d]}{[A + (\frac{n - 1}{2}) D]} = \frac{3 n + 4}{5 n + 6} - - (1)$

Now we have to find

$\frac{a + 6 d}{A + 6 D}$

Hence

$\Rightarrow \frac{n - 1}{2} = 6 \Rightarrow n - 1 = 12 \Rightarrow n = 13$

Now putting $n = 13$ in the equation $(1)$

$\Rightarrow \frac{[a + (\frac{13 - 1}{2}) d]}{[A + (\frac{13 - 1}{2}) D]} = \frac{3 \times 13 + 4}{5 \times 13 + 6} \Rightarrow \frac{[a + (\frac{12}{2}) d]}{[A + (\frac{12}{2}) D]} = \frac{43}{71} \Rightarrow \frac{a + 6 d}{A + 6 D} = \frac{43}{71}$

Hence, the ratio of their $7 t h$ term is $43 : 71$

The ratio of the sum of $n$ terms of two A.P.s is $(3 n + 4) : (5 n + 6)$ . Find the ratio of their $7 t h$ term.

Important Questions on Arithmetic Progressions

Learn and Prepare for any exam you want !

The ratio of the sum of n terms of two A.P.s is (3n+4):(5n+6). Find the ratio of their 7th term.

Important Questions on Arithmetic Progressions

Learn and Prepare for any exam you want !

The ratio of the sum of $n$ terms of two A.P.s is $(3 n + 4) : (5 n + 6)$ . Find the ratio of their $7 t h$ term.