Use mathematical induction to prove that the statement is tr

Use mathematical induction to prove that the statement is true for every positive integer n 1.10+2.11+3.12+ + n(n + 9)= n(n + 1)(n + 14) nin+1n+14) What is the first step in a mathematical induction proof? O A. Show that Sk is true. B. Show that S1 is true O C. Show that S., is true D Show that So is true show that 1.10+2.11+3.12 + n(n + 1)(n + 14) :for n write the statements, nin 00+ 14) for n- 1. Wite the statement S +n(n + 9) 1-10- (Type your answer in factored form.) Simplify S1 on the right 10 ls S1 a true statement? O Yes No

Solution

multiple questions posted. please post each question seperately

1)

B.show that S₁ is true

1*10 =1(1+1)(1+14)_/3

10=10

yes S1 is a true statement

assume statement is true for S_k

1*10 +2*11 +3*12 +......+k(k+9)=k(k+1)(k+14)_/3

now for n=k+1

S_k+1=1*10 +2*11 +3*12 +......+k(k+9) +(k+1)(k+1+9)

S_k+1=(k(k+1)(k+14)_/3)+((k+1)(k+1+9))

S_k+1=(k(k+1)(k+14)_/3)+((k+1)(k+10))

S_k+1=(k+1)(k(k+14)+3(k+10))_/3

S_k+1=(k+1)(k²+17k+30)_/3

S_k+1=(k+1)(k+2)(k+15)_/3

S_k+1=(k+1)((k+1)+1)((k+1)+14)_/3

so statement is true for S_k+1

so by mathematical induction  1*10 +2*11 +3*12 +......+n(n+9)=n(n+1)(n+14)_/3

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2)

first term ,a_o=48_/(1/2)=96

second term,a₁=48

third term,a₂=48*(1_/2) =24

fourth  term,a₃=48*(1_/2)² =12

fifth term,a₄=48*(1_/2)³ =6