1. ## 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

2. Hello, rpatel!

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: ( $a$ = first term, $d$ = common difference)

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

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

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

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

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

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

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

3. thank you Soroban

much appreichated

thanks again

regards