Sequences help

• Nov 8th 2006, 08:23 AM
rpatel
Sequences help
hello

i need some help with sequences. i have been faced with a question which i haven't come accross before. i have no examples to this type of question aswel so i really need some help

the question is :

The sum of the first six terms of an artithmetic series is 21, and the seventh term is three times the sum of the third and fourth.
Find the first erm and the common difference.

i will appreichate any help

thank you

kind regards

:)
• Nov 8th 2006, 09:00 AM
Soroban
Hello, rpatel!

Quote:

The sum of the first six terms of an artithmetic series is 21.
The seventh term is three times the sum of the third and fourth.
Find the first term and the common difference.

You are expected to know these formulas: ($\displaystyle a$ = first term, $\displaystyle d$ = common difference)

The $\displaystyle n^{th}$ term: .$\displaystyle a_n\:=\:a + (n-1)d$

The sum of the first $\displaystyle n$ terms: .$\displaystyle S_n \:=\:\frac{n}{2}\left[2a + (n-1)d\right]$

We are told that sum of the first six terms is 21: .$\displaystyle S_6 \:=\:\frac{6}{2}\left[2a + 5d\right] \:=\:21$
. . which simplifies to: .$\displaystyle 2a + 5d\:=\:7$ [1]

The $\displaystyle 7^{th}$ term is: $\displaystyle a_7 \:=\:a + 6d$
The $\displaystyle 3^{rd}$ term is: $\displaystyle a_3 \:=\:a + 2d$
The $\displaystyle 4^{th}$ term is: $\displaystyle a_4\:=\:a + 3d$

We are told: the $\displaystyle 7^{th}$ term is 3 times the sum of the $\displaystyle 3^{rd}$ and $\displaystyle 4^{th}.$
Hence, we have: .$\displaystyle a + 6d \:=\:3\left[(a + 2d) + (a + 3d)\right]$
. . which simplifies to: .$\displaystyle 5a + 9d \:=\:0$ [2]

Solve the system of equations: .$\displaystyle \begin{array}{cc} (1)\\(2)\end{array} \begin{array}{cc}2a + 5d \:= \:7\\ 5a + 9d \:=\:0\end{array}$ .(Use your favorite method)

And get: .$\displaystyle \boxed{a = -9,\;d = 5}$

• Nov 8th 2006, 09:36 AM
rpatel
thank you Soroban

much appreichated

thanks again

regards

:)