Number of terms in the expansion of (1+x)^2013

To find the number of terms in the expansion of (1+x)^2013, we can use the Binomial Theorem. The Binomial Theorem states that for any positive integer n:

(1+x)^n = C(n,0) + C(n,1)x + C(n,2)x^2 + ... + C(n,n-1)x^(n-1) + C(n,n)x^n,

where C(n,k) represents the binomial coefficient, also known as "n choose k", which is the number of ways to choose k items from a set of n items.

Understanding the Binomial Theorem

The Binomial Theorem is based on the concept of expanding a binomial expression raised to a positive integer power. In the case of (1+x)^n, each term in the expansion is obtained by multiplying the appropriate binomial coefficient with the corresponding powers of 1 and x.

Applying the Binomial Theorem for (1+x)^2013

In our case, we are interested in finding the number of terms in the expansion of (1+x)^2013. Using the Binomial Theorem, we know that there will be a total of 2014 terms in the expansion.

Here's why:

- The exponent of x starts from 0 and goes up to n, which is 2013 in our case. Therefore, there are 2014 different powers of x in the expansion.

- Each power of x corresponds to a term in the expansion, and since there are 2014 different powers of x, there will be 2014 terms in the expansion.

Conclusion

The number of terms in the expansion of (1+x)^2013 is 2014. This can be determined using the Binomial Theorem, which tells us that the number of terms is equal to the exponent of x plus one. In this case, the exponent of x is 2013, so there will be 2014 terms in the expansion.