Question

There are 300 butterfly eggs on a leaf. 10 eggs hatch on the first day. the number of eggs that hatch increases by 25 each day. how many days will it take for all the eggs to hatch? explain your answer.

Answer 1

By sum of the nth term of the arithmetic sequence we can say that, it will take total 5 number of days to hatch all the eggs.

Let the number of days taken by butterfly to hatch the eggs be n.

According to the given question.

Total number of butterfly eggs = 300

Number of eggs hatch on first day = 10

The number of eggs that hatch increases by 25 each day.

Therefore,

Number of eggs hatch on second day = 10 + 25 = 35

Number of eggs hatch on third day = 35 + 25 = 60

Number of eggs hatch on fourth day = 60 + 25 = 85

Like that we have arithmetic sequence with a common difference of 25.

10, 35, 60, 85, ...

Here, it is given that the total number of eggs are 300.

Therefore,

300 = (n/2)[2(10) + (n -1)25]

⇒ 600 = n[20 + 25n -25]

⇒ 600 = n[25n -5]

⇒ 600 = 5n[5n - 1]

⇒ 120 = n[5n -1]

⇒ 120 = 5n^2 - n

⇒ 5n^2 - n - 120 = 0

$n = \frac{1\pm\sqrt{1^{2} -4(5)(120)} }{10}$

⇒ $n = \frac{1\pm \sqrt{2401} }{10}$

⇒ n = 1 ± 49/10

⇒ n = 50/10 0r -48/10 (not possible)

⇒ n = 5

Hence, the total number of days according to arithmetic sequence by the butterfly to hatch the eggs is 5 .

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