An arithmetic series contains n terms. show that if t1 = a−b and tn = a+b then the value of sn is independent of b.

[arithmetic sequences]

Sn is independent of b

Step-by-step explanation:

t1=a-b

tn=a+b

we know that nth term of arithmetic series is  an=a1+(n-1)d

so

a+b=a-b+(n-1)d

⇒2b=(n-1)d-----------equation 1

formula for sum of n terms of arithmetic series

  Sn=(2a1+(n-1)d)

   ⇒Sn=(2(a-b)+2b)      (since (n-1)d=2b  from equation 1)

   ⇒Sn=(2a)  

therefore we can see that Sn is independent of b

hello,

b) 62.47% of the students are between 14 and 18 years old.

step-by-step explanation:

mean = 15 and standard deviation = 2

student between 14 to 18

formula used z-score

for age 14 , z score

using z-table , p(z> -0.5)=0.3085

for age 18 , z score

using z-table , p(z< 1.5)=0.9332

p(-0.5< z< 1.5)=0.9332-0.3085 = 0.6247

thus, 62.47% of the students are between 14 and 18 years old.

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step-by-step explanation: