Question

Sn = a + (a + d)+(a + 2d) + ... + (ℓ − 2d)+(ℓ − d) + ℓ. 2 n(a + ℓ) . We have found the sum of an arithmetic progression in terms of its first and last terms, a and ℓ, and the number of terms n.​

Answer 1

Answer:

1. Sequences

What is a sequence? It is a set of numbers which are written in some particular order. For

example, take the numbers

1, 3, 5, 7, 9, . . . .

Here, we seem to have a rule. We have a sequence of odd numbers. To put this another way, we

start with the number 1, which is an odd number, and then each successive number is obtained

by adding 2 to give the next odd number.

Here is another sequence:

1, 4, 9, 16, 25, . . . .

This is the sequence of square numbers. And this sequence,

1, −1, 1, −1, 1, −1, . . . ,

is a sequence of numbers alternating between 1 and −1. In each case, the dots written at the

end indicate that we must consider the sequence as an infinite sequence, so that it goes on for

ever.

On the other hand, we can also have finite sequences. The numbers

1, 3, 5, 9

form a finite sequence containing just four numbers. The numbers

1, 4, 9, 16

also form a finite sequence. And so do these, the numbers

1, 2, 3, 4, 5, 6, . . ., n .

These are the numbers we use for counting, and we have included n of them. Here, the dots

indicate that we have not written all the numbers down explicitly. The n after the dots tells us

that this is a finite sequence, and that the last number is n.

Here is a sequence that you might recognise:

1, 1, 2, 3, 5, 8, . . . .

This is an infinite sequence where each term (from the third term onwards) is obtained by adding

together the two previous terms. This is called the Fibonacci sequence.

We often use an algebraic notation for sequences. We might call the first term in a sequence

u1, the second term u2, and so on. With this same notation, we would write un to represent the

n-th term in the sequence. So

u1, u2, u3, . . ., un

would represent a finite sequence containing n terms. As another example, we could use this

notation to represent the rule for the Fibonacci sequence. We would write

un = un−1 + un−2

to say that each term was the sum of the two preceding terms.

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