# Find first term and common difference of Arithmetic Sequence, given two other terms

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!

HINT....The nth term of an arithmetic sequence is $$a_n=a+(n-1)d$$ Therefore you have to solve simultaneously $$a+6d=7b+5c$$ $$a+10d=11b+9c$$

• I got the right answer. Thank you. However, can you tell me what you think about my reasoning before 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". Does this have to be true for all the terms? What are your thoughts on this? – Nikos Aug 11 '15 at 20:17
• What you suggest would be true if $a+(n-1)d=(n+2)b+nc, \forall n$ Which would lead to the requirement that $d=b+c$ and $a=3b+c$ – David Quinn Aug 11 '15 at 20:26

$$\frac{T_7-T_1}{T_{11}-T_7}=\frac64$$ Substituting values of $T_7, T_{11}$ and solving gives $T_1=b-c$ directly.

• This is like the "distance" of the sub-indexes from the 7th to the 1st term is 6, and from the 11th to the 7th is 4? Does this logically stand true? hmm – Nikos Aug 12 '15 at 15:40
• yes, that's correct...you can check it algebraically. – hypergeometric Aug 12 '15 at 15:55

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 $d = b + c$.

• Yes this is exactly the answer i wanted to see. This is what i found too. But it's wrong! You see this problem is from a book i'm studying and it has the answer, which is a = b - c and d = b + c. I didn't want to mention it at first because i expected an answer like this. Maybe i should have mentioned it.. – Nikos Aug 11 '15 at 19:57
• @RestlessC0bra If the first term of an arithmetic progression is $a$ the seventh term is $a+6d$ - using $a+7d$ puts you a term out. – Mark Bennet Aug 11 '15 at 20:22