Name: Finding N Arithmetic Means Between Two Numbers
Uploaded: 2019-07-19
Duration: 3 min 37 s
Description: When three numbers are in A.P., the middle term is called the arithmetic mean of the other two numbers. Watch this video for complete information.

Question

The number of terms in an

A . P

is even, the sum of odd terms is

24

and the sum of even terms is

30

. If the last term exceeds the first term by

\frac{21}{2}

,find the number of terms in the

A . P

.

Accepted Answer

Let $a$ be the first term and $d$ be the common difference of $A . P$ .

Let the number of terms of the given $A . P$ be $2 n$ .

Hence there are $n$ even terms and $n$ odd terms according to given condition, we get

$a_{1} + a_{3} + a_{5} \dots \dots n terms = 24$

$\Rightarrow a + a + 2 d + a + 4 d \dots . . . n terms = 24$

We know that, sum of $n$ terms of an $A . P .$ is $S_{n} = [2 a + (n - 1) d]$ where $a$ is the first term and $d$ is the common difference.

$\Rightarrow \frac{n}{2} [2 a + (n - 1) 2 d] = 24$

$\Rightarrow n [a + (n - 1) d] = 24$ (i)

Also, $a_{2} + a_{4} + a_{6} \dots n terms = 30$

We know that, $n^{t h}$ term of an $A . P .$ is $a_{n} = a + (n - 1) d$

$\Rightarrow (a + d) + (a + 3 d) + (a + 5 d) \dots n terms = 30$

We know that, sum of $n$ terms of an $A . P .$ is $S_{n} = [2 a + (n - 1) d]$ where $a$ is the first term and $d$ is the common difference.

$∴ \frac{n}{2} [2 (a + d) + (n - 1) 2 d] = 30$

$\Rightarrow n [a + d + (n - 1) d] = 30$

$\Rightarrow n [a + n d] = 30$ (ii)

Also, $a_{2 n} = a_{1} + \frac{21}{2}$

$\Rightarrow a + (2 n - 1) d = a + \frac{21}{2}$

$\Rightarrow (2 n - 1) d = \frac{21}{2}$ (iii)

Consider, equation (i) $-$ equation (ii)

$[n (n - 1) - n^{2}] d = 24 - 30$

$\Rightarrow n d = 6$

On dividing equation (iii) by equation (iv); we get

$\frac{2 n - 1}{n} = \frac{21}{12}$

$\Rightarrow \frac{2 n - 1}{n} = \frac{7}{4}$

$\Rightarrow 8 n - 4 - 7 n \Rightarrow n = 4$

Hence, the required number of terms of given $A . P$ be $2 n$ $i.e. 2 \times 4 i.e. 8$ .

The number of terms in an $A . P$ is even, the sum of odd terms is $24$ and the sum of even terms is $30$ . If the last term exceeds the first term by $\frac{21}{2}$ ,find the number of terms in the $A . P$ .

Important Questions on Sequences and Series

Learn and Prepare for any exam you want !

The number of terms in an A.P is even, the sum of odd terms is 24 and the sum of even terms is 30. If the last term exceeds the first term by 212,find the number of terms in the A.P.

Important Questions on Sequences and Series

Learn and Prepare for any exam you want !

The number of terms in an $A . P$ is even, the sum of odd terms is $24$ and the sum of even terms is $30$ . If the last term exceeds the first term by $\frac{21}{2}$ ,find the number of terms in the $A . P$ .