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Zorluk: OrtaArithmetic and Geometric Progressions (AP and GP)

The sum of the first nn terms of an arithmetic progression is given by Sn=2n2+3nS_n = 2n^2 + 3n. What is the value of the 7th7^{\text{th}} term of the progression?

Cevap: 29

Cevap

The 7th7^{\text{th}} term of the arithmetic progression is 2929.
For any sequence, the nn-th term is found using Tn=Sn−Sn−1T_n = S_n - S_{n-1}. Substituting n=7n = 7 gives S7=2(7)2+3(7)=119S_7 = 2(7)^2 + 3(7) = 119 and S6=2(6)2+3(6)=90S_6 = 2(6)^2 + 3(6) = 90. Thus, T7=119−90=29T_7 = 119 - 90 = 29.

Adım Adım Çözüm

1
State the relationship between the nn-th term TnT_n and the sum of first nn terms SnS_n
Tn=Sn−Sn−1T_n = S_n - S_{n-1}
The sum of the first nn terms minus the sum of the first n−1n-1 terms equals the nn-th term.
2
Calculate the sum of the first 7 terms (S7S_7)
S7=2(7)2+3(7)=119S_7 = 2(7)^2 + 3(7) = 119
Substitute n=7n = 7 into the sum formula Sn=2n2+3nS_n = 2n^2 + 3n.
3
Calculate the sum of the first 6 terms (S6S_6)
S6=2(6)2+3(6)=90S_6 = 2(6)^2 + 3(6) = 90
Substitute n=6n = 6 into the sum formula Sn=2n2+3nS_n = 2n^2 + 3n.
4
Compute the 7th7^{\text{th}} term (T7T_7)
T7=119−90=29T_7 = 119 - 90 = 29
Subtract S6S_6 from S7S_7.

Anahtar Kavram

Relationship between the nth term and the sum of first n terms of an AP
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Konular

Arithmetic and Geometric Progressions (AP and GP)

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