MP Board · Class 10 · Mathematics · Arithmetic ProgressionsIn an arithmetic progression, the 11th term is 16 and the 21st term is 30. Find the common difference \(d\), the first term \(a\), and the sum of the first 15 terms \(S{15}\). Show all steps clearly.

Step 1: Write the nth‑term formula\nThe nth term of an AP is (a_n = a + (n-1)d).

Step 2: Set up equations for the given terms

• For the 11th term: (a_{11} = a + 10d = 16)  (1)
• For the 21st term: (a_{21} = a + 20d = 30)  (2)

Step 3: Eliminate (a) to find (d)\nSubtract (1) from (2):

[ (a + 20d) - (a + 10d) = 30 - 16 ] [ 10d = 14 ] [ d = \frac{14}{10} = 1.4 ]

Step 4: Find the first term (a)\nSubstitute (d = 1.4) into equation (1):

[ a + 10(1.4) = 16 ] [ a + 14 = 16 ] [ a = 2 ]

Step 5: Compute the sum of the first 15 terms\nThe sum formula is (S_n = \frac{n}{2}[2a + (n-1)d]).\nTake (n = 15), (a = 2), (d = 1.4):

[ S_{15} = \frac{15}{2}[2(2) + (15-1)(1.4)] ] [ = 7.5[4 + 14(1.4)] ] [ = 7.5[4 + 19.6] ] [ = 7.5 \times 23.6 ] [ = 177 ]

Final Results

• Common difference (d = 1.4)
• First term (a = 2)
• Sum of first 15 terms (S_{15} = 177)