The 7th and 11th terms of an arithmetic sequence are 7b + 5c and 11b + 9c respectively.
Given these i want to find the first term (term when n = 0) and the common difference. I tried a lot of techniques to solve this, but i end up in logical fallacies. The one thing i surely notice in this though is that the c coefficient is less than 2 from the b coefficient and that the coefficient of b agrees with the number of the term n. So i assume that this is the case with all the terms (maybe i'm wrong). Given the general form of the arithmetic sequence \begin{equation} a_{n} = a + nd \end{equation} how can i find a and d?
Could someone help, or give me a hint? Thanks in advance!
$\endgroup$ 03 Answers
$\begingroup$HINT....The nth term of an arithmetic sequence is $$a_n=a+(n-1)d$$ Therefore you have to solve simultaneously $$a+6d=7b+5c$$ $$a+10d=11b+9c$$
$\endgroup$ 2 $\begingroup$$$\frac{T_7-T_1}{T_{11}-T_7}=\frac64$$ Substituting values of $T_7, T_{11}$ and solving gives $T_1=b-c$ directly.
$\endgroup$ 2 $\begingroup$You know that $a_7 = 7b + 5c = a + 7d$ and $a_{11} = 11b + 9c = a + 11d$. Solving this system of equations, you will get $a = -2c$ and $d = b + c$.
$\endgroup$ 2
