If (3x − 1)7  = a7x7 + a6x6 + a5x5 +...+ a1x + a0, then a7 + a5+ ...+a1 + a0 =

To find the value of a7 + a5 + ... + a1 + a0, we need to understand the given equation (3x - 1)7 = a7x7 + a6x6 + a5x5 + ... + a1x + a0.

Step 1: Expand the left side of the equation using the binomial theorem. The binomial theorem states that (a + b)n = an + (nC1)an-1b + (nC2)an-2b2 + ... + (nCn-1)abn-1 + bn, where nCk represents the binomial coefficient.

In our equation, (3x - 1)7 can be expanded as follows: (3x - 1)7 = (7C0)(3x)7(-1)0 + (7C1)(3x)6(-1)1 + (7C2)(3x)5(-1)2 + ... + (7C6)(3x)(-1)6 + (7C7)(-1)7

Step 2: Simplify the expanded equation. The binomial coefficients (nCk) can be calculated using the formula n! / (k!(n-k)!), where n! represents the factorial of n.

Simplifying the expanded equation, we get: (3x - 1)7 = 1(3x)7 + 7(3x)6(-1) + 21(3x)5(-1)2 + 35(3x)4(-1)3 + 35(3x)3(-1)4 + 21(3x)2(-1)5 + 7(3x)(-1)6 + (-1)7

Step 3: Combine like terms. We can group the terms with the same power of x together.

The expanded equation becomes: (3x - 1)7 = 2187x7 - 5103x6 + 5103x5 - 2835x4 + 945x3 - 189x2 + 21x - 1

Step 4: Compare the expanded equation with the given equation a7x7 + a6x6 + a5x5 + ... + a1x + a0. We can equate the coefficients of the corresponding powers of x.

Comparing the coefficients, we have: a7 = 2187 a6 = -5103 a5 = 5103 a4 = -2835 a3 = 945 a2 = -189 a1 = 21 a0 = -1

Step 5: Calculate the sum of a7 + a5 + ... + a1 + a0. a7 + a5 + ... + a1 + a0 = 2187 + 5103 + 5103 + 2835 + 945 + 189 + 21 + (-1)

Finally, we can simplify the expression to find the sum: a7 + a5 + ... + a1 + a0 = 15782