Series and Sequences (1 Viewer)

horndog huang 2 · Jan 4, 2016

Prove that the sum Sn of the ﬁrst n terms is always 7 less than the ( n +1)th term

The sum Sn of the first n terms : 7(2^n-1)

help pwease
thank you

InteGrand · Jan 4, 2016

horndog huang 2 said:
Prove that the sum Sn of the ﬁrst n terms is always 7 less than the ( n +1)th term

The sum Sn of the first n terms : 7(2^n-1)

help pwease
thank you

$The $(n+1)^{\text{th}}$ term $t_{n+1}$ is$

$\begin{align*}t_{n+1} &= S_{n+1}-S_n \\ &= 7(2^{n+1}-1) - 7(2^n-1) \\ &= 7 \times 2^{n+1} - 7\times 2^n \\ &= 7\times 2^n \left(2-1 \right) \\ &=7\times 2^n . \end{align*}$

$Hence$

$\begin{align*}S_n &=7\left(2^n-1\right) \\ &= 7\times 2^n - 7 \\ &= t_{n+1}-7, \end{align*}$

$as claimed. $\blacksquare$$

horndog huang 2 · Jan 4, 2016

Thanks again integrand!

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Series and Sequences (1 Viewer)

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