For the A.P. : 10, 15, 20, ………, 195; find : i) the number of terms in the above A. P. ii) the sum of all its terms
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Answer:
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Required Solution :-
Here we've been given with an A.P. and firstly we need to calculate the number of terms in it !
As we can clearly notice that we have first term (a) , common difference (d) and the last term (l) of the A.P.
We have :
- First term (a) = 10
- Common difference (d) = 5
- Last term (l) = 195
Remember that in an A.P. if we have last term (l), common difference (d) , and it has n terms then we use the given formula :
- l = a + (n - 1) d
Substituting the values,
→ 195 = 10 + (n - 1) 5
→ 195 = 10 + 5n - 5
→ 195 = 5 + 5n
→ 5n = 195 - 5
→ 5n = 190
→ n = 190 / 5
→ n = 38
Therefore, the given A.P. has 38 terms !
Sum of all the terms :
For an A.P. if , first term (a) , no of terms (n) , and common difference (d) are known then inorder to calculate its sum we use the given formula :
- S = n/2 [2a + (n - 1) d]
Substituting the values,
→ S = 38 / 2 [2(10) + (38 - 1) 5]
→ S = 19 [2(10) + (38 - 1) 5]
→ S = 19 [2 × 10 + (38 - 1) 5]
→ S = 19 [20 + (38 - 1) 5]
→ S = 19 [20 + (37) 5]
→ S = 19 [20 + 37 × 5]
→ S = 19 [20 + 185]
→ S = 19 [205]
→ S = 19 × 205
→ S = 3895
Therefore, sum of the A.P. is 3895