The third term of an A.P is 4m - 2n. If the ninth term of the progression is 2m - 8n. Find the common difference in terms of m a
nd n
1 answer:
Let
denote the <em>n</em>-th term in the progression. So
![a_n=a_{n-1}+d](https://tex.z-dn.net/?f=a_n%3Da_%7Bn-1%7D%2Bd)
for some constant difference between terms <em>d</em>.
Solve for
explicitly:
![a_4=a_3+d](https://tex.z-dn.net/?f=a_4%3Da_3%2Bd)
![a_5=a_4+d=a_3+2d](https://tex.z-dn.net/?f=a_5%3Da_4%2Bd%3Da_3%2B2d)
![a_6=a_5+d=a_3+3d](https://tex.z-dn.net/?f=a_6%3Da_5%2Bd%3Da_3%2B3d)
and so on, up to
![a_n=a_3+(n-3)d](https://tex.z-dn.net/?f=a_n%3Da_3%2B%28n-3%29d)
We're told that the third term is
, and the ninth term is
, and according to the recursive rule above, we have
![a_9=a_3+6d](https://tex.z-dn.net/?f=a_9%3Da_3%2B6d)
Solve for <em>d</em> :
![2m-8n=(4m-2n)+6d](https://tex.z-dn.net/?f=2m-8n%3D%284m-2n%29%2B6d)
![-2m-6n=6d](https://tex.z-dn.net/?f=-2m-6n%3D6d)
![d=-\dfrac{2m+6n}6=\boxed{-\dfrac{m+3n}3}](https://tex.z-dn.net/?f=d%3D-%5Cdfrac%7B2m%2B6n%7D6%3D%5Cboxed%7B-%5Cdfrac%7Bm%2B3n%7D3%7D)
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