(1%2Ba2)...(1%2Ban)-1%2Ba1%2Ba2%2B...%2Ban.jpeg "(1+a1)(1+a2)...(1+an) 1+a1+a2+...+an")
Expansion of (1+a1)(1+a2)...(1+an) and its Relationship with 1+a1+a2+...+an
In algebra, the expansion of the product of binomials of the form (1+a1)(1+a2)...(1+an) is a fundamental concept. In this article, we will explore the expansion of this product and its interesting relationship with the sum 1+a1+a2+...+an.
The Expansion of (1+a1)(1+a2)...(1+an)
Using the distributive property of multiplication over addition, we can expand the product of two binomials as follows:
(1+a1)(1+a2) = 1 + a1 + a2 + a1a2
Now, let's expand the product of three binomials:
(1+a1)(1+a2)(1+a3) = (1 + a1 + a2 + a1a2)(1+a3)
= 1 + a1 + a2 + a3 + a1a2 + a1a3 + a2a3 + a1a2a3
Notice a pattern emerging? We can generalize this expansion to the product of n binomials:
(1+a1)(1+a2)...(1+an) = 1 + a1 + a2 + ... + an + a1a2 + a1a3 + ... + a1an + a2a3 + ... + a2an + ... + an-1an + ... + a1a2...an
The Relationship with 1+a1+a2+...+an
Now, let's examine the sum 1+a1+a2+...+an. We can rewrite this sum as:
1 + a1 + a2 + ... + an = (1 + a1) + a2 + ... + an
= (1+a1)(1) + a2 + ... + an
= (1+a1)(1+a2) + ... + an
= (1+a1)(1+a2)...(1+an) - a1a2 - a1a3 - ... - a1an - a2a3 - ... - a2an - ... - an-1an + ... + a1a2...an
Comparing this with the expansion of (1+a1)(1+a2)...(1+an), we notice that the sum 1+a1+a2+...+an is equal to the product (1+a1)(1+a2)...(1+an) minus all the terms that contain more than one ai factor.
Conclusion
In conclusion, the expansion of (1+a1)(1+a2)...(1+an) is a fascinating concept in algebra that has a beautiful relationship with the sum 1+a1+a2+...+an. By expanding the product of binomials, we can reveal a hidden structure that connects these two seemingly unrelated expressions.
