The 16th term of an arithmetic progression is 53 and the 20th term is 65. find the sum of all the odd numbers in the sequence that are less than 100 please show with steps

Sum = 795

Step-by-step explanation:

an = a + (n - 1)*d

16th Term

a_16 = a + (16 - 1) * d

a_16 = a + 15 * d

53 = a + 15 d

a_20 = a + 19*d

65 = a + 19d

53 = a + 15d           Subtract

12 = 4d                   Divide by 4

3 = d

53 = a + 15d

53 = a + 15*3

53 = a + 45

a = 53 - 45

a = 8

The first odd term in the sequence is 8 + 3 = 11

The next term is 11 + 3 = 14

The third term is 17

So the difference is 17 - 11 = 6

The last term is

L = a + (n - 1)*6

L = 11 + (n - 1)*6

I think we have to guess at what the last term is. Let n = 10

L = 11 + 9*6    It's not large enough

Try 15

L = 11 + 14*6

L = 11 + 84

L = 95         That's correct. The last term in the sequence is 95

So  

The sum is

Sum = (a + L)*n/2

n = 15

Sum = (11 + 95)*15/2

Sum = (106)*15/2

Sum = 53*15/2

Sum = 795

29.5977692753

step-by-step explanation:

11.2

step-by-step explanation:

when doing tangent the tangent of the angle is equal to the opposite side from the angle over the adjacent side to the angle.

tan43 = x / 12

move 12 over by multiplying each side by 12

12tan43 = x

x = about 11.2