Question

Raju, a Grade-10 student once visited a botanical garden with his classmates as a part of the field trip organised in his school. He observed a unique pattern in the arrangement of petals in Dahlia flower. He observed that there were 4 petals in the innermost whorl, 7 petals in the second whorl, 10 in the third whorl and so on.. The flower has 25 whorls in total. Help Raju with his following doubts.
How many petals will be there in the 20th whorl?

Answer 1

Answer:

There will be 61 petals in the 20th whorl.

Step-by-step-explanation:

NOTE: Kindly refer to the attachment first.

The number of petals increases by 3 in two consecutive whorls.

There are 4 petals in the first whorl or the innermost whorl.

$\therefore\sf\:a\:=\:4$

And the difference between the number of petals in two consecutive whorls is 3.

$\because\sf\:t_{2}\:-\:t_{1}\:=\:7\:-\:4\:=\:3\\\\\sf\:t_{3}\:-\:t_{2}\:=\:10\:-\:7\:=\:3$

$\therefore\sf\:d\:=\:3$

This is an A.P.

There are total 25 whorls.

$\therefore\sf\:n\:=\:25$

So, we have to find the total number of petals in 20th whorl.

It means, we have to find $\sf\:t_{n}$.

$\boxed{\red{\sf\:t_{n}\:=\:a\:+\:(\:n\:-\:1\:)\:d}}\\\\\implies\sf\:t_{20}\:=\:4\:+\:(\:20\:-\:1\:)\:\times\:3\\\\\implies\sf\:t_{20}\:=\:4\:+\:19\:\times\:3\\\\\implies\sf\:t_{20}\:=\:4\:+\:57\\\\\implies\boxed{\red{\sf\:t_{20}\:=\:61}}$

Ans.: There will be 61 petals in the 20th whorl.

Additional Information:

1. Arithmetic Progression:

In a sequence, if the common difference between two consecutive terms is constant, then the sequence is called as Arithmetic Progression ( A. P. ).

2. $\sf\:n^{th}$ term of an A. P. :

The number of a term in the given A. P. is called $\sf\:n^{th}$ term of A. P.

3. Formula for $\sf\:n^{th}$ term of an A. P. :

$\boxed{\red{\sf\:t_{n}\: =\:a\:+\:(\:n\:-\:1\:)\:d}}$

4. The sum of the first n terms of an A. P. :

The addition of either all the terms or a particular $\sf\:n^{th}$ terms is known as sum of n terms of A. P.

5. Formula for sum of the first n terms of A. P. :

$\boxed{\red{\sf\:S_{n}\:=\:\frac{n}{2}\:[\:2a\:+\:(\:n\:-\:1\:)\:d\:]}}$