Matematika · Problems · Teori Bilangan

Soal Faktorial

Tentukan jumlah dari  \sum\limits_{k=1}^{n}k.k!=1.1!+2.2!+...+n.n! !

Penyelesaian:

Perhatikan bahwa

\begin{aligned} k.k!&=(k+1-1).k! \\  &= (k+1)k!-k! \\ &= (k+1)!-k! \end{aligned}

Diperoleh

\begin{aligned} \sum\limits_{k=1}^{n}k.k! &= 1.1!+2.2!+...+n.n! \\ &=(2! -1!)+(3!-2!)+ \ldots +((n+1)!-n!) \\&=(n+1)!-1 \end{aligned}

Catatan:

Dengan menggunakan induksi matematika dapat ditunjukkan bahwa

\sum\limits_{k=1}^{n}k.k!=(n+1)!-1.

 

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.