The sum of the first 7 terms of an AP is 49 and yhe sum of its first 17 terms is 289 . find he sum of its first n term (10th class question)
Answers
Answer - n raised to the power 2
Step-by-step explanation:
Hope this helps u
GIVEn
The sum of the first 7 terms of an AP is 49 and the sum of its first 17 terms is 289 .
TO FINd
find he sum of its first n term
SOLUTIOn
- S₇ = 49
- S₁₇ = 289
→ S₇ = n/2[2a + (n - 1)d]
→ 49 = 7/2[2a + (7 - 1)d]
→ 49 = 7/2[2a +6d]
→ 49 × 2/7 = [2a + 6d]
→ 2a + 6d = 14
→ 2(a + 3d) = 14
→ a + 3d = 7 ---(i)
_____________________
→ S₁₇ = n/2[2a + (n - 1)d]
→289 = 17/2[2a + (17-1)d]
→ 289 × 2/17 = 2a + 16d
→ 2a + 16d = 34
→ 2(a + 8d) = 34
→ a + 8d = 17 ---(ii)
_____________________
Subtract both the equations
→ (a + 3d) - (a + 8d) = 7 - 17
→ a + 3d - a - 8d = -10
→ - 5d = -10
→ d = 10/5 = 2
Putting the value of d in eqⁿ - (i)
→ a + 3d = 7
→ a + 3 × 2 = 7
→ a + 6 = 7
→ a = 7 - 6 = 1
Hence, a = 7 and d = 2
_____________________
Now, the value of sum of " n " terms
→ Sn = n/2[2a + (n - 1)d]
→ Sn = n/2[2×1 + (n - 1)2]
→ Sn = n/2[2 + 2n - 2]
→ Sn = n/2 × 2n
→ Sn = n²