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Improved Euler method with Butcher Tableau

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Use the order conditions to prove that the improved Euler method with Butcher Tableau

0 |
1 | 1
---+-------------
| 1/2 1/2

is second order but not third order.

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The expert examines the improved Euler method Butcher Tableau is depicted.

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Problem Statement:

Use the order conditions to prove that the improved Euler method with Butcher Tableau

is second order but not third order.
Order Conditions:
First, let us write out what the order conditions are for a general r-stage Runge-Kutta method. Say we have the following Butcher tableau for our method:
c_1 a_11 〖... a〗_1r

⋮ ⋮ ⋮
c_r a_r1 ...〖 a〗_rr
b_1 ...b_r

To guarantee at least first order accuracy, and in addition ...

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