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3 years ago

The sum of the first 5 terms of an AP is 30 and the sum of the four term from T6 to T9 (inclusive) is 69. Find the AP

Mathematics

1 answer:

Ainat [17]3 years ago

7 0

Answer: The AP = 1, ⁷/₂, 6, ¹⁷/₂, 11 ..............

Step-by-step explanation:

From the first statement,

S₅ = ⁵/₂(2a + ( n - 1 )d } = 30

5(2a + 4d )d = 60

10a + 20d = 60

reduce to lowest term to easy calculation by dividing through by 10

a + 2d = 6 -----------------------------------1

second statement

sum of the next 4 terms inclusive

T₉ = ⁹/₂(2a + 8d ) = 69

9(2a + 8d ) = 30 + 69

18a + 72d = 99 x 2

18a + 72d = 198

divide through by 18 to reduce to lowest time

a + 4d = 11 ------------------------------------------2

Now solve the two equation simultaneously to find a and d

a + 2d = 6

a + 4d = 11

-2d = -5

d = ⁵/₂.

Now substitute for d to get a

a + 2(⁵/₂) = 6

a + 5 = 6

a = 6 - 5

a = 1.

Therefore the AP = 1 , ⁷/₂ , 6 , ¹⁷/₂ , 11 , ..............

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Step-by-step explanation:

Check for conditions

For this case in order to use the normal distribution for this case or the 68-95-99.7% rule we need to satisfy 3 conditions:

a) Independence : we assume that the random sample of 400 students each student is independent from the other.

b) 10% condition: We assume that the sample size on this case 400 is less than 10% of the real population size.

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So then we have all the conditions satisfied.

Solution to the problem

For this case we know that the distribution for the population proportion is given by:

$p \sim N(p, \sqrt{\frac{p(1-p)}{n}})$

So then:

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68% of the rates are expected to be betwen 0.718 and 0.762

$\mu_p -2*\sigma_p = 0.74-2*0.0219=0.696$

$\mu_p +2*\sigma_p = 0.74+2*0.0219=0.784$

95% of the rates are expected to be betwen 0.696 and 0.784

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The sum of the first 5 terms of an AP is 30 and the sum of the four term from T6 to T9 (inclusive) is 69. Find the AP​

The sum of the first 5 terms of an AP is 30 and the sum of the four term from T6 to T9 (inclusive) is 69. Find the AP