Question Video: Writing an Expression for the Sum of an Arithmetic Sequence | Nagwa Question Video: Writing an Expression for the Sum of an Arithmetic Sequence | Nagwa

# Question Video: Writing an Expression for the Sum of an Arithmetic Sequence Mathematics • Second Year of Secondary School

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Find an expression for the sum of an arithmetic sequence whose first term is π and whose common difference is π.

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### Video Transcript

Find an expression for the sum of an arithmetic sequence whose first term is π and whose common difference is π.

We are told in the question that the first term of our arithmetic sequence is π and the common difference is π. We are trying to find an expression for the sum of the first π terms which we will write as π sub π. This will be equal to the first term π plus the second term π plus π and so on. We know that the πth term of any arithmetic sequence is equal to π plus π minus one multiplied by π. This means that the penultimate term is equal to π plus π minus two multiplied by π. We will call this equation one.

We will then reverse the order of this sum, which we can do as addition is commutative. This gives us π sub π is equal to π plus π minus one multiplied by π plus π plus π minus two multiplied by π and so on and finally plus π plus π plus π. We will call this equation two. Adding equation one and equation two gives us two multiplied by π sub π on the left-hand side. On the right-hand side, we will add each pair of terms. Adding the first pair, we see that π plus π is equal to two π. So we have two π plus π minus one multiplied by π.

The second pair of terms have the same sum as π plus π is equal to two π and π plus π minus two π is equal to π minus one π. In fact, this will be true for each of the pairs in our equations. Each pair of terms will sum to give us two π plus π minus one multiplied by π. We have π of these terms, so we can rewrite the right-hand side as π multiplied by two π plus π minus one multiplied by π. Dividing both sides of our equation by two gives us π sub π is equal to π over two multiplied by two π plus π minus one multiplied by π. This is an expression for the sum of an arithmetic sequence whose first term is π and whose common difference is π.

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