If
r=1∑3030Cr−1r3(30Cr)2=α×229
, then
α
is equal to:
Get the complete, step-by-step math solution for: "If _{r=1}^{30} {r³ (^{30}C_r)²}{^{30}C_{r-1}} = α × 2^{29} , then α is equal to:". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Simplify the ratio of binomial coefficients
We begin by simplifying the ratio of the binomial coefficients
30Cr−130Cr
. Using the property
nCr−1nCr=rn−r+1
, we substitute
n=30
to get
r30−r+1
, which simplifies to
r31−r
.
Step 2: Substitute the simplified ratio into the sum
Now, we substitute the simplified ratio back into the original summation. The term
30Cr−1(30Cr)2
can be rewritten as
30Cr−130Cr×30Cr
. This allows us to replace the ratio with the expression we just found.
Step 3: Simplify the term inside the summation
We can simplify the term inside the summation further. One
r
from
r3
cancels with the
r
in the denominator of
r31−r
, leaving us with
r2(31−r)30Cr
.
Step 4: Expand and use the identity
r⋅nCr=n⋅n−1Cr−1
Expand the term inside the summation to get
31r230Cr−r330Cr
. We will evaluate each part separately. We use the identity
r⋅nCr=n⋅n−1Cr−1
to simplify terms involving
r⋅30Cr
.
Step 5: Evaluate the sums using known identities
We use the identities
r=1∑nr2⋅nCr=n(n−1)2n−2+n2n−1
and
r=1∑nr3⋅nCr=n22n−1+n(n−1)2n−2
. Substituting
n=30
into these identities gives the values for the sums.
Step 6: Substitute the evaluated sums back into the expression
Now, we substitute the calculated values of the two summations back into the expression
31r=1∑30r230Cr−r=1∑30r330Cr
.
Step 7: Calculate the final value and find
α
Factor out
30×228
and perform the arithmetic. We then adjust the power of 2 to match the form
α×229
to find the value of
α
.