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Question: How do you find the sum of the arithmetic sequence given \({{a}_{1}}=7\) , \(d=-3\) , \(n=20\) ?...

How do you find the sum of the arithmetic sequence given a1=7{{a}_{1}}=7 , d=−3d=-3 , n=20n=20 ?

Explanation

Solution

Before starting the problem, we first write down the expression for the nth{{n}^{th}} term. Then, putting the values in it, we get the nth{{n}^{th}} term. The average will be the algebraic mean of the nth{{n}^{th}} and the first terms. The sum will be the average times the number of terms.

Complete step by step solution:
A sequence is an enumerated collection of objects, especially numbers, in which repetitions are allowed and in which the order of objects matters. A sequence may be finite or infinite depending on the number of objects in the sequence. Sequences can be of various types such as arithmetic sequence, geometric sequence and so on. Sequences can be completely random as well. The nth{{n}^{th}} term of a sequence is sometimes written as a function of n.
An arithmetic sequence is the one in which the difference between the consecutive objects is a constant. It is called the common difference of the arithmetic sequence. The general expression of the nth{{n}^{th}} term will be,
an=a+(n−1)d....(1){{a}_{n}}=a+\left( n-1 \right)d....\left( 1 \right) where a is the first term of the sequence and d is the common difference.
The values of a, d and n are given as a1=7{{a}_{1}}=7 , d=−3d=-3 , n=20n=20 . We now put these values in the above equation and get,
an=7+(20−1)(−3) ⇒an=−50 \begin{aligned} & {{a}_{n}}=7+\left( 20-1 \right)\left( -3 \right) \\\ & \Rightarrow {{a}_{n}}=-50 \\\ \end{aligned}
Now, we know that since the terms of an arithmetic sequence are arranged linearly, the average of the sequence will be the algebraic mean of the terms, i.e.,
average=a+an2 ⇒average=7+(−50)2 ⇒average=21.5 \begin{aligned} & average=\dfrac{a+{{a}_{n}}}{2} \\\ & \Rightarrow average=\dfrac{7+\left( -50 \right)}{2} \\\ & \Rightarrow average=21.5 \\\ \end{aligned}
The sum of the sequence will be the average times the number of terms which means,
sum=average×n ⇒sum=(−21.5)×20 ⇒sum=−430 \begin{aligned} & sum=average\times n \\\ & \Rightarrow sum=\left( -21.5 \right)\times 20 \\\ & \Rightarrow sum=-430 \\\ \end{aligned}

Therefore, we can conclude that the sum of the sequence is −430-430.

Note: This problem was quite easy, but we should be careful in the calculations. This problem can also be done in another way. There is a predefined formula for the sum of an arithmetic sequence which is sum=n2[2a+(n−1)d]sum=\dfrac{n}{2}\left[ 2a+\left( n-1 \right)d \right] . If we put the given values in this formula, we directly get the sum.