Question 3 Find the sum to n terms of each of the series in Exercises 1 to 7.
3 × 12 + 5 × 22 + 7 × 32 + …
Class X1 - Maths -Sequences and Series Page 196
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Let S = 3 × 1² + 5 × 2² + 7 × 3² +.......
we observed from this series,
Tn = (nth term of 3, 5, 7.....)(nth term of 1,2,3..)²
={3 + (n-1)×2}{1+(n-1)×1}²
= (2n + 1)n²
= 2n³ + n²
S =∑Tn
= ∑(2n³ + n²)
= 2∑n³ + ∑n²
= 2{n(n+1)/2}² + n(n+1)(2n+1)/6
=n(n+1)/2[2 × n(n+1)/2 + (2n+1)/3]
= n(n+1)/2[{3n(n+1) + 2n+1}/3]
=n(n+1)(3n²+5n+1)/6
we observed from this series,
Tn = (nth term of 3, 5, 7.....)(nth term of 1,2,3..)²
={3 + (n-1)×2}{1+(n-1)×1}²
= (2n + 1)n²
= 2n³ + n²
S =∑Tn
= ∑(2n³ + n²)
= 2∑n³ + ∑n²
= 2{n(n+1)/2}² + n(n+1)(2n+1)/6
=n(n+1)/2[2 × n(n+1)/2 + (2n+1)/3]
= n(n+1)/2[{3n(n+1) + 2n+1}/3]
=n(n+1)(3n²+5n+1)/6
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