The following problem refers to an arithmetic sequence If th

The following problem refers to an arithmetic sequence. If the fifth term is 17 and the tenth term is 37, find the term a1, the common difference d, and then find a23 and S23.

a1=

d=

a23=

s23=

Solution

Given that we have an AP sequence such that 5th term a5 = 17, 10th term a10 = 37

We know a5 = a+4d = 17...........equation 1

also a10 = a+9d = 37.............. equation 2

Solving equations 1 and 2

Equation 2 - equation 1 results in

a+9d-a-4d =37-17

=> 5d = 20

=> d = 20/5 = 4

Thereffore common difference d = 4

Substituting value of d = 4 in equation 1

a+4d = 17

=> a + 4*4 = 17

=> a +16 = 17

=> a = 17 - 16 = 1

Therefore first term a = a1 = 1

Now a23 = a+22d = 1+22*4 = 1+88 = 89

Therefore a23 = 89

s23 = (n/2)(2a+(n-1)d) = (23/2)(2*1+(23-1)*4) = (23/2)(2+22*4) = (23/2)(2+88) = (23/2)(90) =23*45 =1035

Thus Sum of 1st 23 terms S23 = 1035