StudentAidUnit 8: Mathematical Inductions and Binomial Theorem — Exercise 8 1
11th Class Mathematics · Unit 8: Mathematical Inductions and Binomial Theorem
8.Aliza invests Rs. 1,000,000 in a business that promises a 6% return compounded annually. Prove by mathematical induction that amount of money after n years is 1,000,000(1.06)^n
Given
Initial investment
Rs. 1,000,000
Annual rate
6% per annum
Formula
Formula to prove
S(n) = 1,000,000(1.06)n
Working
Basic case (n=1)
S(1) = 1,000,000(1.06)1
= 1,000,000 × 1.06
Result
Amount after 1 year
= 1,060,000
10.Sikander save Rs. 500 in the first month and increases his savings Rs. 500 each month. Using mathematical induction, determine whether his total savings will reach at least Rs. 12,000 after 24 months.
Given
First month saving
a = 500
Monthly increment
d = 500
Formula
Sum of arithmetic series
S(n) = n2[2a + (n-1)d]
Working
Substitute given values
S(n) = n2[1000 + (n-1)500]
= 250n(n+1)
Verification: n = 24
S(24) = 250(24)(25)
= 6000 × 25
Result
Total savings after 24 months
= 150,000
11.Prove by mathematical induction that if Ali takes a loan of Rs. 2,000,000 and pay Rs. 50,000 at the end of each year, the remaining balance after n years is R_n = 2,000,000 - 50,000n
Given
Initial loan
Rs. 2,000,000
Annual payment
Rs. 50,000
Formula
Remaining balance after n years
Rn = 2,000,000 - 50,000n
Working
Basic case (n=1)
R1 = 2,000,000 - 50,000(1)
= 2,000,000 - 50,000
Result
Remaining balance after 1 year
= 1,950,000
12.If Salman start savings with Rs. 5,000 and saves an additional Rs. 1,000 at the end of every month, derive a formula S(n) for his total saving after n months. Prove the correctness of the formula using mathematical induction.
Given
Initial saving
Rs. 5,000
Monthly saving
Rs. 1,000
Formula
Total savings after n months
S(n) = 5000 + 1000n
Working
Basic case (n=1)
S(1) = 5000 + 1000(1)
= 5000 + 1000
Result
Total savings after 1 month
= 6000