A Series huh!

Algebra Level 5

Consider a series :

$1,7,28,84,210,\dots$

Let $T_n$ be its $n^{th}$ term & $S_n$ be the sum of its first $n$ terms.

Find the value of $\dfrac{S_{30}}{T_{20}}$ .

Give your answer correct upto three places of decimal.

Try more from my set Algebra Problems .

The answer is 47.135.

1 solution

Purushottam Abhisheikh
Mar 4, 2015

The Given Series is :

$^6C_6, ^7C_6, ^8C_6 \dots$

So, $T_{20} = ^{25}C_6 =177100$

Now the real problem is calculating $S_{30}$ .

$S_{30} = ^6C_6+ ^7C_6+ ^8C_6 + \dots + ^{35}C_6$

$= ^7C_7+ ^7C_6+ ^8C_6 + \dots + ^{35}C_6$

$= ^8C_7 + ^8C_6 + ^9C_6 \dots + ^{35}C_6$ $(\because ^nC_{r+1} +^nC_r = ^{n+1}C_{r+1})$

$= ^9C_7 +^9C_6 + ^{10}C_6 + \dots + ^{35}C_6$

$\vdots$

$= ^{35}C_7 + ^{35}C_6$

$= ^{36}C_7 = 8347680$

Hence the answer is 47.1354

Hi , I thought so too about the sequence . But the first 5 terms of this sequence are 0 !! Are you allowed to manipulate the sequence ? Btw here is the link

A Former Brilliant Member - 6 years, 3 months ago

Is it not allowed?

Purushottam Abhisheikh - 6 years, 3 months ago

I don't know , but since I am the only one who has this doubt , let's leave it as it is :)

A Former Brilliant Member - 6 years, 3 months ago

@A Former Brilliant Member – Well, the series of natural number is $1,2,3,4,5,...$ . Now, can I not start it with 3, as in the series can be $3,4,5,6,7,8,...$ ?

Pranjal Jain - 6 years, 3 months ago

@Pranjal Jain – Yes , I had thought so too, but since I got one try wrong , I thought that I should verify it with the question maker .

A Former Brilliant Member - 6 years, 3 months ago

It would be better to post how you added the terms, I mean writing up the formulae you have used.

Pranjal Jain - 6 years, 3 months ago

@Pranjal Jain – Ok. I shall edit it.

Purushottam Abhisheikh - 6 years, 3 months ago

@Purushottam Abhisheikh – Looks better! Thanks $\ddot\smile$

Pranjal Jain - 6 years, 3 months ago

A Series huh!

The answer is 47.135.

1 solution

0 pending reports