1. Convert the following  into meters.

       a. 260 cm      b.8900 cm     c.2.3 km

          a. 260 cm       =  260 ×  ¹/₁₀₀ m   =  ²⁶⁰/₁₀₀ m   = 2.60 m

          b. 8900 cm    =  8900 × ¹/₁₀₀ m  = 89 m

          c. 2.3 km        =    2.3 ×   1000 m = 2300 m.

2. Convert the following into kilometers.

       a. 2000 m        b. 200 m     c. 100  m

:  a. 2000 m      = 2000 × ¹/₁₀₀₀ km   = 2 km

          b. 200 m       = 200 × ¹/₁₀₀₀ km   = 0. 2 km

          c. 100 m       = 100 × ¹/₁₀₀₀ km    = 0. 1 km.

3. Convert the following into kilograms.

       a. 2 tonne     b.  7 .2 tonne       c.  2800 g.

       a.    2 tonne    = 2 × 1000   kg      = 2000 kg

      b.    7.2 tonne  = 7.2 × 1000  kg    = 7200 kg

      c.      2800 g    = 2800 ×¹/₁₀₀₀ kg   = 2.8 kg

4. Convert the following to degrees Fahrenheit.

       a. 20^oc       b. 80^oc.

                  T_f  =  ⁹/₅ T_c  + 32  

       a. 20^oc   =  ⁹/₅  × 20 + 32     =   36 + 32  = 68^oF

      b.  80^oc  = ⁹/₅ ×80 + 32       =     (9 × 16 +   32)^o F

                                                     = (144 + 32)^o F   = 176^oF

5. Convert the following to degrees celcius.

       a. 50^o F     b. 275^oF

                          T_e    =   ⁵/₉ (T_f - 32)

              a.  50^oF      =  (50 - 32)   = ⁵/₉ × 18   = 10^oC

              b. 275^of      =   ⁵/₉ (275 - 32)^oC  = 5/9  (243)^oC.

                                  =  (5 × 27)^oC  = 135^oC.

6. Convert the following to kelvin.

      a.  30^oc   b. 120^oc

               T_k    = T_c    + 273

   a.  30^oc      = (30 + 273) K

                       = 303 K

  b.  -120^oC  = (-120 + 273)K

                      = 153 K.

7. If u_n +1  = u_n + 5  and u₁ =3, find u₂, u₃ and u₄ and state whether or not the sequence is arithmetic.

           Given that u₁ = 3

             u_n + 1  = U_n = 5

             u₁   = 3

             u₂  = u₁ + 5   = 3 + 5  = 8

             u₃  = u₂ + 5   = 8 + 5  = 13

              u₄ = u₃ + 5   = 13 + 5 = 18

              u₂ - u₁  = 8 - 3 = 5

              u₃ - u₂  = 13 - 8 = 5

              u₄ - u₃  = 18 - 13 = 5

  so, this is an arithmetic sequence.

8. For the sequence 5, 8, 11, 14, 17.

        i) . deduce the formula for the n^th term

        ii) . calculate the 10^th term.

          Position  : 1                  2                 3                  4              5

          Term       :  5                 8                11                14            17

                             ↓                 ↓                  ↓                 ↓                ↓

                        3× 1 + 2      3 × 2 + 2     3× 3+2      3× 4+2     3 × 5 + 2

       i) so nth term    = 3 × n + 2   = 3n + 2

     ii)  10^th term   = 3 × 10 + 2 = 30 + 2  = 32.

9. For the sequence 5, 9, 13, 17, 21.

       i. deduce the common difference d

     ii. give the formula for the nth term

    iii. Calculate the 50th term.

     i. Common difference    = 9 - 5 = 4

   ii. Position  :     1                  2                 3                  4              5               n

          Term       :  5                 9                13                17            21

                             ↓                 ↓                  ↓                 ↓                ↓                  ↓

                        4× 1 + 1      4 × 2 + 1     4× 3+1      4× 4+1     4 × 5 + 1     4 × n + 1   

             so nth term  = 4n + 1

         ii.   50 th term   = 4  50 + 1

                                  = 200 + 1  = 201

10. Find the sum of 3 + 8  + 13 + ..... + 53

            u_n  = 53    and d = 8 - 3 = 5  and u₁  = 3

            u₁ + (n - 1) 5  = 53

              3 + (n-1) 5   = 53  

              3 + (n-1) 5   = 53          (n-1) 5  = 53 - 3  = 50

                                                     n -1       = ⁵⁰/₅      = 10

                                                     n  =  10 + 1 = 11

        S_n   = ⁿ/₂  (u₁ + u_n)     =  ¹¹/₂   (3 + 53)

                                             =  ¹¹/₂  ×  56  = ⁶¹⁶/₂  = 308

11. Evaluate   (4 - n).

          u₁  = 4 - 1  = 3

          u₁₀ = 4 - 10  = -6

        Sn     =  ⁿ/₂   (u₁ + u_n)

         S₁₀  = ¹⁰/₂  (3 + -6)  = 5 × -3 = -15.