the fifth term of an ap is 18 and the 9th term is 30 find the sum of the first 25 terms

Answer: 1050

Step-by-step solution

Step 1: Set up equations using the nthn\text{th} term formula

We know that the nthn\text{th} term of an AP is given by an=a+(n−1)da_n = a + (n - 1)d, where aa is the first term and dd is the common difference. For the 5th5\text{th} term, a5=a+(5−1)d=a+4d=18a_5 = a + (5 - 1)d = a + 4d = 18, which we label as equation (1). For the 9th9\text{th} term, a9=a+(9−1)d=a+8d=30a_9 = a + (9 - 1)d = a + 8d = 30, which we label as equation (2).

Step 2: Solve for the common difference dd and first term aa

Subtracting equation (1) from equation (2) eliminates aa, giving 4d=124d = 12, which simplifies to d=3d = 3. Substituting d=3d = 3 into equation (1) gives a+4(3)=18a + 4(3) = 18, so a+12=18a + 12 = 18, which yields a=6a = 6.

Step 3: Calculate the sum of the first 2525 terms

We use the sum formula Sn=n2[2a+(n−1)d]S_n = \frac{n}{2}[2a + (n - 1)d] with n=25n = 25, a=6a = 6, and d=3d = 3. Substituting these values, we get S25=252[12+24×3]=252(84)=25×42=1050S_{25} = \frac{25}{2}[12 + 24 \times 3] = \frac{25}{2}(84) = 25 \times 42 = 1050.

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