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Question Number 208218 by hardmath last updated on 07/Jun/24

$${\mathrm{a}}_{\mathbf{n}}\mathrm{numbers}\mathrm{series}$$$$\mathrm{If}{\mathrm{S}}_{16}-{\mathrm{S}}_{13}={\mathrm{S}}_{106}-{\mathrm{S}}_{103}$$$$\mathrm{Find}:\frac{{3\mathrm{a}}_3+{4\mathrm{a}}_4+{5\mathrm{a}}_5}{{2\mathrm{a}}_{12}}=?$$

Commented by mr W last updated on 07/Jun/24

$$thisisnotthefirsttimewheniask$$$$youwhatyoumeanwith‘‘numbers$$$${series}^{″},e.g.inQ207930.$$

Commented by hardmath last updated on 07/Jun/24

$$$$For example, there is a numerical series and a geometric series

Commented by mr W last updated on 08/Jun/24

$$thenyoumeanarithmeticseries.$$

Commented by hardmath last updated on 08/Jun/24

$$\mathrm{yes}\mathrm{dear}\mathrm{professor}$$

Answered by zamin2001 last updated on 07/Jun/24

$$S_{16}-S_{13}=a_{16}+a_{15}+a_{14}=3a_1+42d★$$$$S_{106}-S_{103}=a_{106}+a_{105}+a_{104}=3a_1+312d◼$$$$★=◼\Rightarrow d=0\Rightarrow constantnumericalseries$$$$\frac{{3\mathrm{a}}_3+{4\mathrm{a}}_4+{5\mathrm{a}}_5}{{2\mathrm{a}}_{12}}=\frac{3C+4C+5C}{2C}=\frac{12C}{2C}=6$$

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