Answer:
[tex]\frak{Given}\begin{cases} \sf{\: Common \: difference \: (d) = 7}\\\sf{ \: 22nd \ term = 149} \end{cases}[/tex]
We've to find out the sum of 20 terms. So, n = 22
By using nth term Formula of the AP :
[tex]\star \ \boxed{\sf{\purple{a_{n} = a + (n -1)d}}} \\ \\ \underline{\bf{\dag} \: \mathfrak{Substituting \ Values \ in \ the \ formula \ :}}[/tex]
[tex]:\implies\sf 149 = a + (22 - 1) \times 7 \\\\\\:\implies\sf 149 = a + 21 \times 7 \\\\\\:\implies\sf 149 = a + 147\\\\\\:\implies\sf a = 149 - 147\\\\\\:\implies\boxed{\frak{\purple{a = 2}}}[/tex]
[tex]\therefore\underline{\textsf{ Here, we get value of the First term (a) of AP \textbf{2}}}. \\[/tex]
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For any Arithmetic Progression ( AP ), the sum of n terms is Given by :
[tex]\bf{\dag}\quad\large\boxed{\sf S_n = \dfrac{n}{2}\bigg \lgroup a + l\bigg \rgroup}[/tex]
Where :
- a = First Term
- n = no. of terms
- l = Last Term
[tex]:\implies\sf S_{22} = \dfrac{\cancel{20}}{\cancel{\:2}} \bigg(2 + 149 \bigg) \\\\\\:\implies\sf S_{22} = 10 \times 151 \\\\\\:\implies\boxed{\frak{\purple{ S_{22} = 1510}}}[/tex]
[tex]\therefore\underline{\textsf{ Hence, Sum of 20 terms of the AP is \textbf{1510}}}. \\[/tex]
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