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satela [25.4K]
3 years ago
13

In arithmetic sequence tn find: S10, if t2=5 and t8=15

Mathematics
1 answer:
makkiz [27]3 years ago
7 0

Given:

\bold{t_2=5}\\\\\bold{t_8=15}\\\\

To find:

\bold{T_n=?}\\\\\bold{S_{10}=?}

Solution:

\to \bold{t_2=5}\\\\  \bold{a+d=5} \\\\  \bold{a=5-d} ................(i)\\\\ \to \bold{t_8=15}\\\\ \bold{a+ 7d=15}.................(ii)\\\\

Putting the equation (i) value into equation (ii):

\to \bold{5-d+7d=15}\\\\\to \bold{5+6d=15}\\\\\to \bold{6d=15-5}\\\\\to \bold{6d=10}\\\\\to \bold{d=\frac{10}{6}}\\\\\to \bold{d=\frac{5}{3}}\\\\

Putting the value of (d) into equation (i):

\to \bold{a=5-\frac{5}{3}}\\\\\to \bold{a=\frac{15-5}{3}}\\\\\to \bold{a=\frac{10}{3}}\\\\

Calculating the \bold{t_n :}

\to \bold{t_n=a+(n-1)d}\\\\\to \bold{t_n=\frac{10}{3}+(n-1)\frac{5}{3}}\\\\\to \bold{t_n=\frac{10}{3}+\frac{5}{3}n-\frac{5}{3}}\\\\\to \bold{t_n=\frac{10}{3}-\frac{5}{3} +\frac{5}{3}n}\\\\\to \bold{t_n=\frac{10-5}{3} +\frac{5}{3}n}\\\\\to \bold{t_n=\frac{5}{3} +\frac{5}{3}n}\\\\ \to \bold{t_n=\frac{5}{3}(1+n)}\\\\

Formula:

\bold{S_n = \frac{n}{2}[2a + (n - 1)d ] }\\

Calculating the \bold{S_{10}} :

\to \bold{S_{10} = \frac{10}{2}[2\frac{10}{3} + (10- 1)\frac{5}{3} ] }\\\\\to \bold{S_{10} = 5[\frac{20}{3} + (9)\frac{5}{3} ] }\\\\\to \bold{S_{10} = 5[\frac{20}{3} + 9 \times \frac{5}{3} ] }\\\\\to \bold{S_{10} = 5[\frac{20}{3} + 3\times 5 ] }\\\\\to \bold{S_{10} = 5[\frac{20}{3} + 15 ] }\\\\\to \bold{S_{10} = 5[\frac{20+ 45}{3} ] }\\\\\to \bold{S_{10} = 5[\frac{65}{3} ] }\\\\\to \bold{S_{10} = 5 \times \frac{65}{3}  }\\\\\to \bold{S_{10} =  \frac{325}{3}  }\\\\\to \bold{S_{10} =108.33 }

Therefore, the final answer is  " \bold{\frac{5}{3}(1+n)\ and\ 108.33}"

Learn more:

brainly.com/question/11853909

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