Name: Sum of First n Terms of an Arithmetic Progression
Uploaded: 2019-07-19
Duration: 7 min 10 s
Description: 13+13+13+13…….How do we find the sum of 50 terms of this series? Learn to find the sum of the first ‘n’ terms of an Arithmetic Progression with a real-life e...

Question

A farmer plants apple trees in a square pattern. In order to protect the apple trees against the wind he plants conifer trees all around the orchard.

Here you see a diagram of this situation where you can see the pattern of apple trees and conifer trees for any number ( $n$ ) of rows of apple trees:

Complete the table:

$n$	Number of apple trees	Number of conifer trees
$3$	$9$	$24$
$4$	$16$	$32$
$5$
$6$
$7$

Accepted Answer

We can see that, in first figure, there is one apple tree and $8$ conifer trees.

In second figure, there are four trees and $16$ conifer trees.

In third figure, there are nine apple trees and $24$ conifer trees.

In fourth figure, we have $16$ apple trees and $32$ conifer trees.

So, we can see the number of apple trees forms a pattern i.e., $1^{2}, 2^{2}, 3^{2}, 4^{2} . . . . .$

Number of apple trees in $n$ rows $=$ $n^{2}$

Number of apple trees in $5$ rows $=$ $5^{2} = 25$ .

Number of apple trees in $6$ rows $= 6^{2} = 36$ .

And also number of apple trees in $7$ rows $= 7^{2} = 49$

As the number of conifer trees increases i.e., $8, 16, 24, . . . . . . .$ etc

So, we have the first term $= 8$ , the second term $= 16$ and third term $=$ $24$

Therefore, $a_{1} = 8, a_{2} = 16, a_{3} = 24, . . . . . .$

Now, the differences between two consecutive terms are,

$d_{1} = a_{2} - a_{1} = 16 - 8 = 8$

$d_{2} = a_{3} - a_{2} = 24 - 16 = 8$

Since, $d_{1} = d_{2}$ i.e., the difference between two consecutive terms is constant i.e., $d = 8$

Therefore, $8, 16, 24, . . . . . . .$ is forms an A.P.

We know that, the $n^{th}$ term of an A.P, $a_{n} = a + (n - 1) d$ , where $a =$ the first term and $d =$ common difference.

For $n = 5$ , $a_{5} = 8 + (5 - 1) \times 8 = 8 + 32 = 40$

For $n = 6$ , $a_{6} = 8 + (6 - 1) \times 8 = 8 + 40 = 48$

For $n = 7$ , $a_{7} = 8 + (7 - 1) \times 8 = 8 + 48 = 56$

Therefore, the completed table is,

$n$ Number of apple trees Number of conifer trees

$3$ $9$ $24$

$4$ $16$ $32$

$5$ $25$ $40$

$6$ $36$ $48$

$7$ $49$ $56$

Important Questions on Arithmetic Progressions

Learn and Prepare for any exam you want !

Number of tables	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
Number of seats	$n^{th}$	$6$	$8$