Prove or disprove: for any odd integer x, (x^2 -1) is divisible by 8.

True

Step-by-step explanation:

To prove by induction, first show that the statement is true at an initial value (in this case, x = 1).

8 | (1² − 1)

8 | 0

Next, assume the statement is true at x = k.

8 | (k² − 1)

Now show that it is true at the next value of x (x = k + 2).

(k + 2)² − 1

= k² + 4k + 4 − 1

= k² − 1 + 4(k + 1)

k + 1 is an even number, so 4(k + 1) is a multiple of 8.  And since we're assuming that k² − 1 is a multiple of 8, that means the sum is also a multiple of 8.

8 | ((k + 2)² − 1)

So the statement is true.

Logically, here's another way of proving it:

x² − 1 = (x − 1) (x + 1)

Since x is an odd integer, both x − 1 and x + 1 are even numbers.  Since the difference between them is 2, one of them is a multiple of 4.  An even number times a multiple of 4 is a multiple of 8.

plants store their excess carbohydrate in the form of starch so that it could be used as energy source later on. the excess carbohydrate is converted into starch in the plastid i.e. chloroplast & this process occurs in the presence of sunlight. it is an insoluble polymer of glucose which gets stored in cell organelles named as amyloplasts. in some plants, starch is also stored in roots and embryos.

y + 11 = (1/4)(x + 16)

step-by-step explanation: