Mathematics: What is a1 of the arithmetic sequence for which a3=126 and a64=3,725 a 64 = 3 , 725 ?

hello! your answer is 4. I just completed the assignment and got it correct. Hope it helps....

What is a1 of the arithmetic sequence for which

stodd9503

20.08.2021

In an arithmetic sequence, every pair of consecutive terms differs by a fixed number c, so that the n-th term a_n is given recursively by

$a_n=a_{n-1}+c$

Then for n ≥ 2, we have

a_2=a_1+c

a_3=a_2+c = (a_1+c)+c = a_1 + 2c

a_4=a_3+c = (a_1 + 2c) + c = a_1 + 3c

and so on, up to

a_n=a_1+(n-1)c

Given that a_3=126 and $a_{64}=3725$ , we can solve for a_1 :

$\begin{cases}a_1+2c=126\\a_1+63c=3725\end{cases}$

$\implies(a_1+63c)-(a_1+2c)=3725-126$

$\implies 61c = 3599$

$\implies c=59$

$\implies a_1+2\times59=126$

$\implies a_1+118 = 126$

$\implies \boxed{a_1=8}$

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