Hi Angel.
An arithmetic sequence is one where the difference between successive terms is a common difference (let's call it C). That is,
That means that when you compare U_{20} to U_{6}, you get:
From this expression, you can figure out what C is. Then you can use the same idea in reverse to figure out the first term U_{1}:
Once you know the common difference and the first element in the arithmetic sequence, you can represent the sum of 28 terms as:
Which you can expand to:
This looks like a lot of adding to figure out, but look what happens if you write it out twice: once forward and once backward, then add the two equations:
You get all the same terms! Since there are 28 of them (from 0C to 27C), you can simplify this to:
Now you just have to solve for S, plug in the numbers you got for U_{1} and C and you have your sum.
Now in part (b) of your question, you have two different sums and you have to express the second sum in sigma notation. You'll have to use the first two sums to find U_{1} and C.
Let's look at the general form of the sum:
You can write out both of your sums S_{10} and S_{15} using 10 and 15 as the values of n. Then you have two equations with two unknowns (U_{1} and C) that you can solve in the usual way. Once you have U_{1} and C, you can substitute those into the form of the sigma notation I showed you earlier to finish your answer.
Hope this helps!
Sue
