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## Homework Statement

The average number of mRNAs in the cell at any time t is <m>(t) = Σ m * p(t). Sum over all the differential equations derived in a) in order to obtain a differential equation for <m>(t)

## Homework Equations

So the differential equation I got in a) was dp/dt = (-k

_{p}* P

_{m}) - (m * k

_{m}* P

_{m}) + (k

_{p}* P

_{m-1}) + ((m+1) * k

_{m}* P

_{m+1})

That would make d<m>/dt = Σ m * dP

_{m}/dt = Σ m * ((-k

_{p}* P

_{m}) - (m * k

_{m}* P

_{m}) + (k

_{p}* P

_{m-1}) + ((m+1) * k

_{m}* P

_{m+1}))

What I need is d<m>/dt without any p(t)'s left on the right hand side

## The Attempt at a Solution

I first separated the k

_{p}and k

_{m}terms:

d<m>/dt = Σ m * k

_{p}* (-P

_{m}+ P

_{m-1}) + Σ m * k

_{m}* ((-m * P

_{m}) + ((m+1) * P

_{m+1}))

Now I know I have to manipulate this using substitutions such as m' = m -1, m = (m-1) + 1, and shifting of indices, but I am completely unsure how to proceed. Any help would be greatly appreciated.

Sorry I forgot to mention, the summations are from m = 0 to infinity

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