Discussion

Explanation:

Let the 3^rd term of AP = T₃ = a [Here, a is not the first term of the AP]

14^th term of AP = T₁₄ = ar and
69^th term of AP = T₆₉ = ar² 

Now, T₁₄ - T₃ = 11d [d is the common difference of the AP]
Also, T₆₉ - T₁₄ = 55d

⇒ 5(T₁₄ - T₃) = T₆₉ - T₁₄ 
⇒ 5(ar - a) = ar² - ar
⇒ r² - 6r + 5 = 0
⇒ (r - 1)(r - 5) = 0
⇒ r = 1 or 5
[r = 1 is rejected as the given terms are distinct.]

∴ T₃ = a
T₁₄ = 5a
T₆₉ = 25a

⇒ d = (5a - a)/11 = 4a/11

Now, next term of geometric progression will be 125a = T₃ + (n - 3)d
⇒ 125a = a + (n - 3) × 4a/11
⇒ 341 = n - 3
⇒ n = 344.

∴ The next term of the geometric progression is the 344^th term of the GP.

Hence, 344.

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