(a) Simplify; \(\frac{log_2 ^8 + log_2 ^{16} - 4 log_2 ^2}{log_4^{16}}\) (b) The first, third, and seventh terms of an Arithmetic Progression (A.P) from thr...

(a) Simplify; \(\frac{log_2 ^8 + log_2 ^{16} - 4 log_2 ^2}{log_4^{16}}\)

(b) The first, third, and seventh terms of an Arithmetic Progression (A.P) from three consecutive terms of a Geometric Progression (G.P). If the sum of the first two terms of the A.P is 6, find its:

(I) first term; (ii) common difference.

Answer Details

(a) To simplify the expression, we can use the logarithmic properties:

log_a^m + log_aⁿ = log_a^m*n

log_a^m - log_aⁿ = log_a^m/n

First, we simplify the numerator:

log₂⁸ + log₂¹⁶ - 4 log₂² = log₂^8*16 - 4 log₂²² = log₂¹²⁸ - 4 log₂⁴ = 7 - 4 * 2 = 3

Next, we simplify the denominator:

log₄¹⁶ = log₂²⁴ = 4

Finally, we divide the numerator by the denominator:

3 / 4 = 0.75

So the simplified expression is 0.75.

(b) Given the first, third, and seventh terms of an Arithmetic Progression (A.P), we can find the common difference "d" and the first term "a" using the formula for the nth term of an A.P:

a_n = a + (n-1)d

Let's call the first term "a", the common difference "d", and the number of terms "n".

The third term is:

a + 2d

The seventh term is:

a + 6d

We know that the sum of the first two terms is 6, so we can write an equation using the formula for the nth term:

a + (a + d) = 6

2a + d = 6

2a = 6 - d

2a = 6 - d

a = (6 - d) / 2

We can substitute "a" in one of the other equations to find "d":

a + 2d = a + 2(6 - a) / 2 = 7

7 = a + 3 - a = 3

a = 4

So the first term is 4 and the common difference is (6 - a) / 2 = (6 - 4) / 2 = 1.

So the first term of the Arithmetic Progression is 4 and the common difference is 1.