Solveeit Logo
Login

Question

Question: Find the first four terms of an Arithmetic Progression whose first term is \(-2\) and common differe...

Find the first four terms of an Arithmetic Progression whose first term is −2-2 and common difference is −2.-2.

Explanation

Solution

An arithmetic progression has a common difference. If the differences of the consecutive terms are the same, then it is called the common difference of the arithmetic progression. We can find the nth{{n}^{th}} term of the arithmetic progression by an=a1+(n−1)d{{a}_{n}}={{a}_{1}}+\left( n-1 \right)d where an{{a}_{n}} is the nth{{n}^{th}} term, a1{{a}_{1}} is the first term and dd is the common difference.

Complete step by step answer:
We have already learnt that an arithmetic progression is a sequence with the differences of the consecutive terms are the same. And this difference is called the common difference.
Let us suppose that we are given with the first term a1{{a}_{1}} of an arithmetic progression. Also, the common difference dd is also given. Then it is easy to find the nth{{n}^{th}} term an{{a}_{n}} by adding the first term to the product of the common difference and n−1.n-1.
That is, an=a1+(n−1)d.{{a}_{n}}={{a}_{1}}+\left( n-1 \right)d.
We can apply this to find the first four terms of the A.P whose first term is −2-2 and common difference is −2.-2.
So, a1=−2{{a}_{1}}=-2 and d=−2.d=-2.
Let us find the second term using the above formula. So, here n−1=1.n-1=1.
Therefore, a2=−2+1×(−2)=−2+−2=−4.{{a}_{2}}=-2+1\times \left( -2 \right)=-2+-2=-4.
So, we can find the third term similarly by taking n−1=2.n-1=2.
We will get a3=−2+2×(−2)=−2−4=−6.{{a}_{3}}=-2+2\times \left( -2 \right)=-2-4=-6.
Now, the fourth term can be found when we fix n−1=3.n-1=3.
So, a4=−2+3×(−2)=−2−6=−8.{{a}_{4}}=-2+3\times \left( -2 \right)=-2-6=-8.

Hence, the first four terms are −2,−4,−6,−8.-2,-4,-6,-8.

Note: This can be directly done by adding the common difference dd to the first term, the second term and the third term. Since the first term is a1=−2{{a}_{1}}=-2 and d=−2,d=-2, the second term is a2=−2+(−2)=−4.{{a}_{2}}=-2+\left( -2 \right)=-4. The third term is a3=−4+(−2)=−6{{a}_{3}}=-4+\left( -2 \right)=-6 and the fourth term is a4=−6+(−2)=−8.{{a}_{4}}=-6+\left( -2 \right)=-8.