The fifth term of an A.P. is 26 and its 10𝑡ℎ term is 51. Find the A.P.

We have been given the fifth and tenth term of an 𝐴. 𝑃 and we need to find the 𝐴. 𝑃 . 

It is known that the 𝑛𝑡ℎ term of an AP is given by

𝑎_𝑛 = 𝑎 + (𝑛 − 1)𝑑, where 𝑎_𝑛 is the 𝑛^𝑡ℎ term, 𝑎 is the first term, 𝑛 is the term to be found and 𝑑 is the common difference of the A.P.

Therefore, the 5𝑡ℎ term (𝑎5) is given by

𝑎5 = 𝑎 + (5 − 1)𝑑

⇒ 𝑎5 = 𝑎 + 4𝑑

⇒ 26 = 𝑎 + 4𝑑                  − (𝑖)

Similarly, the 10𝑡ℎ term (𝑎10) is given by

𝑎10 = 𝑎 + (10 − 1)𝑑

⇒ 𝑎10 = 𝑎 + 9𝑑

⇒ 51 = 𝑎 + 9𝑑                 − (𝑖𝑖)

Now, we need to solve (𝑖) and (𝑖𝑖) to obtain values of 𝑎 and 𝑑 . 

Subtracting (𝑖𝑖) from (𝑖), 

we get 25 = 5𝑑 ⇒ 𝑑 = 5 Substituting 𝑑 = 5 in equation (𝑖), 

we get 26 = 𝑎 + 4 × 5 

⇒ 𝑎 = 6

Any 𝐴. 𝑃 can be expressed using 𝑎 and 𝑑 as : 

𝑎, 𝑎 + 𝑑, 𝑎 + 2𝑑, 𝑎 + 3𝑑 𝑎𝑛𝑑 𝑠𝑜 𝑜𝑛 

Therefore, the 𝐴. 𝑃. is 6, 6 + 5, 6 + 2 × 5, 6 + 3 × 5 𝑎𝑛𝑑 𝑠𝑜 𝑜𝑛 

Hence, the 𝐴. 𝑃 = 6, 11, 17,..