The-sum-of-the-4-th-and-6-th-terms-of-an-AP-is-42-the-sum-of-the-third-and-9th-terms-of-the-proression-is-52-Find-the-first-term-the-common-difference-and-the-sum-of-the-first-10-terms-of-t

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Question Number 10583 by Saham last updated on 19/Feb/17

$$\mathrm{The}\mathrm{sum}\mathrm{of}\mathrm{the}{4}^{\mathrm{th}}\mathrm{and}{6}^{\mathrm{th}}\mathrm{terms}\mathrm{of}\mathrm{an}\mathrm{AP}\mathrm{is}42.\mathrm{the}\mathrm{sum}\mathrm{of}$$$$\mathrm{the}\mathrm{third}\mathrm{and}9\mathrm{th}\mathrm{terms}\mathrm{of}\mathrm{the}\mathrm{proression}\mathrm{is}52.\mathrm{Find}\mathrm{the}$$$$\mathrm{first}\mathrm{term},\mathrm{the}\mathrm{common}\mathrm{difference}\mathrm{and}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{the}\mathrm{first}$$$$10\mathrm{terms}\mathrm{of}\mathrm{the}\mathrm{progression}.$$

Commented by Saham last updated on 19/Feb/17

$$\mathrm{i}\mathrm{have}\mathrm{reduce}\mathrm{it}\mathrm{sir}.$$

Commented by sandy_suhendra last updated on 19/Feb/17

$$\mathrm{are}\mathrm{you}\mathrm{Tawakalitu}?$$

Commented by Saham last updated on 20/Feb/17

$$\mathrm{yes}\mathrm{sir}.\mathrm{i}\mathrm{used}\mathrm{my}\mathrm{surname}\mathrm{now}.$$

Answered by ajfour last updated on 19/Feb/17

$$2a+8d=42$$$$2a+10d=52$$$$sod=5;a=1$$$$T_1+T_2+\dots +T_{10}=5(2+45)=235.$$$$$$

Answered by mrW1 last updated on 19/Feb/17

$$a_3=a+2d$$$$a_4=a+3d$$$$a_6=a+5d$$$$a_9=a+8d$$$$$$$$a_4+a_6=42$$$$\iff (a+3d)+(a+5d)=42$$$$2a+8d=42$$$$a+4d=21\dots \left(i\right)$$$$$$$$a_3+a_9=52$$$$\iff (a+2d)+(a+8d)=52$$$$2a+10d=52$$$$a+5d=26\dots \left(ii\right)$$$$$$$$\left(ii\right)-\left(i\right):d=5\left(commondifference\right)$$$$a=21-4\times 5=1\left(firstterm\right)$$$$$$$$a_{10}=a+9d=1+9\times 5=46$$$$$$$$S_{10}=\frac{10(1+46)}{2}=235$$

Commented by Saham last updated on 19/Feb/17

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.$$

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