בסדרה חשבונית נתון:
s 10 = 120 , s 15 = 255 {\displaystyle s_{10}=120,s_{15}=255} מצא את a 1 , d {\displaystyle a_{1},d}
S 10 = 10 [ 2 a 1 + ( 9 ) d ] 2 = 120 {\displaystyle S_{10}={\frac {10[2a_{1}+(9)d]}{2}}=120}
S 15 = 15 [ 2 a 1 + 14 d ] 2 = 255 {\displaystyle S_{15}={\frac {15[2a_{1}+14d]}{2}}=255}
נצמצם:
5 [ 2 a 1 + ( 9 ) d ] = 120 {\displaystyle 5[2a_{1}+(9)d]=120}
15 ∗ 2 [ a 1 + 7 d ] 2 = 255 {\displaystyle {\frac {15*2[a_{1}+7d]}{2}}=255}
ועוד פעם:
2 a 1 + ( 9 ) d = 24 {\displaystyle 2a_{1}+(9)d=24}
15 [ a 1 + 7 d ] = 255 {\displaystyle 15[a_{1}+7d]=255}
ופעם נוספת:
2 a 1 + 9 d = 24 {\displaystyle 2a_{1}+9d=24}
a 1 + 7 d = 17 {\displaystyle a_{1}+7d=17}
כלומר a 1 = 17 − 7 d {\displaystyle a_{1}=17-7d}
נציב: 2 ( 17 − 7 d ) + 9 d = 24 {\displaystyle 2(17-7d)+9d=24}
נפתח: 34 − 14 d + 9 d = 24 {\displaystyle 34-14d+9d=24}
נבודד: 5 d = 10 {\displaystyle 5d=10}
נקבל d = 2 {\displaystyle d=2}
נמצא את האיבר ה-1:
a 1 = 17 − 7 ∗ 2 = 3 {\displaystyle a_{1}=17-7*2=3}
![{\displaystyle S_{10}={\frac {10[2a_{1}+(9)d]}{2}}=120}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81a2971a64c0626ca5a12c20cc29b2fbdf4f2620)
![{\displaystyle S_{15}={\frac {15[2a_{1}+14d]}{2}}=255}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d6e76d32f0c27a218ce0592c0ff059751055031)
![{\displaystyle 5[2a_{1}+(9)d]=120}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7ee1991ed22853e89764f7be31e05da1266ab3e)
![{\displaystyle {\frac {15*2[a_{1}+7d]}{2}}=255}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42aef6bbd9098bd666d7f5af33121315035c2afb)
![{\displaystyle 15[a_{1}+7d]=255}](https://wikimedia.org/api/rest_v1/media/math/render/svg/178f91a74f80c1d3b42a80e03f08cc7d2393f0aa)
