Find the sum of all two digit numbers which when divided by 7 yield 1 as the remainder. Solve using arithmetic progression with working.
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⟶ ᴀɴꜱᴡᴇʀ :
⟶ ꜱᴏʟᴜᴛɪᴏɴ :-
ᴛʜᴇ ʀᴇQᴜɪʀᴇᴅ ꜱᴇʀɪᴇꜱ ɪꜱ 15, 22, 29, .,99
ᴛʜᴇꜱᴇ ᴀʀᴇ ɪɴ ᴀᴘ,
ʜᴇʀᴇ, ᴀ = 15, ᴅ = 7
ɴᴏᴡ, ᴀ(ɴ) = ᴀ + (ɴ - 1)ᴅ
→ 99 = 15 + (ɴ - 1)7
→ 84/7 = ɴ - 1
→ ɴ = 12 + 1
→ ɴ = 13
ɴᴏᴡ, ꜱ(ɴ) = ɴ/2(2ᴀ + (ɴ - 1)ᴅ)
⟹ꜱ(ɴ) = 13/2(2 x 15 + (13 - 1)7)
⟹ꜱ(ɴ) = 13/2(30 + 84)
⟹ꜱ(ɴ) = 13/2 x 114
⟹ꜱ(ɴ) = 741
ʜᴇɴᴄᴇ, ᴛʜᴇ ꜱᴜᴍ ᴏꜰ ᴀʟʟ ᴛᴡᴏ ᴅɪɢɪᴛ ɴᴀᴛᴜʀᴀʟ ɴᴜᴍʙᴇʀꜱ ɪꜱ 741.
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